2.1: The Function Game
Each group has three people. Designate one person as the “Leader” and one person as the “Recorder.” (These roles will rotate through all three people.) At any given time, the Leader is looking at a sheet with a list of “functions,” or formulas; the Recorder is looking at the answer sheet. Here’s how it works.
- One of the two players who is not the Leader says a number.
- The Leader does the formula (silently), comes up with another number, and says it.
- The Recorder writes down both numbers, in parentheses, separated by a comma. (Like a point.)
- Keep doing this until someone guesses the formula. (If someone guesses incorrectly, just keep going.)
- The Recorder now writes down the formula—not in words, but as an algebraic function.
- Then, move on to the next function.
Sound confusing? It’s actually pretty easy. Suppose the first formula was “Add five.” One player says “\begin{align*}4\end{align*}” and the Leader says “\begin{align*}9\end{align*}”. One player says “\begin{align*}-2\end{align*}” and the Leader says “\begin{align*}3\end{align*}”. One player says “\begin{align*}0\end{align*}” and the Leader says “\begin{align*}5\end{align*}”. One player says “You’re adding five” and the Leader says “Correct.” At this point, the Recorder has written down the following:
1. Points: \begin{align*}(4,9) (-2,3) (0,5)\end{align*}
Answer: \begin{align*}x+5\end{align*}
Sometimes there is no possible answer for a particular number. For instance, your function is “take the square root” and someone gives you “\begin{align*}-4\end{align*}.” Well, you can’t take the square root of a negative number: \begin{align*}-4\end{align*} is not in your domain, meaning the set of numbers you are allowed to work on. So you respond that “\begin{align*}-4\end{align*} is not in my domain.”
Leader, do not ever give away the answer!!! But everyone, feel free to ask the teacher if you need help.
The Function Game Leader’s Sheet
Only the leader should look at this sheet. Leader, use a separate sheet to cover up all the functions below the one you are doing right now. That way, when the roles rotate, you will only have seen the ones you’ve done.
- Double the number, then add six.
- Add three to the number, then double.
- Multiply the number by \begin{align*}-1\end{align*}, then add three.
- Subtract one from the number. Then, compute one divided by your answer.
- Divide the number by two.
- No matter what number you are given, always answer “\begin{align*}-3\end{align*}.”
- Square the number, then subtract four.
- If you are given a positive number, give the same number back. If you are given a negative number, multiply that number by \begin{align*}-1\end{align*}.
- Cube the number.
- Add two to the number. Also, subtract two from the original number. Multiply these two answers.
- Take the square root of the number. Round up to the nearest integer.
- Add one to the number, then square.
- Square the number, then add \begin{align*}1\end{align*}.
- Cube the number. Then subtract the original number from that answer.
- Give back the lowest prime number that is greater than or equal to the number.
- If you are given an odd number, respond \begin{align*}1\end{align*}. If you are given an even number, respond \begin{align*}2\end{align*}. (Fractions are not in the domain of this function.)
The Function Game: Answer Sheet Recorder
1. Points:
Answer:
2. Points:
Answer:
3. Points:
Answer:
4. Points:
Answer:
5. Points:
Answer:
6. Points:
Answer:
7. Points:
Answer:
8. Points:
Answer:
9. Points:
Answer:
10. Points:
Answer:
11. Points:
Answer:
12. Points:
Answer:
13. Points:
Answer:
14. Points:
Answer:
15. Points:
Answer:
16. Points:
Answer:
Name: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
Homework: The Function Game
1. Describe in words what a variable is, and what a function is.
There are seven functions below (numbered 2-8). For each function,
- Write the same function in algebraic notation.
- Generate three points from that function.
For instance, if the function were “Add five” the algebraic notation would be “\begin{align*}x+5\end{align*}”. The three points might be \begin{align*}(2,7)\end{align*}, \begin{align*}(3,8)\end{align*}, and \begin{align*}(-5,0)\end{align*}.
2. Triple the number, then subtract six.
a. Algebraic notation: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
b. Three points: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
3. Return \begin{align*}4\end{align*}, no matter what.
a. Algebraic notation: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
b. Three points: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. Add one. Then take the square root of the result. Then, divide that result into two.
a. Algebraic notation: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
b. Three points: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
5. Add two to the original number. Subtract two from the original number. Then, multiply those two answers together.
a. Algebraic notation: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
b. Three points: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
6. Subtract two, then triple.
a. Algebraic notation: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
b. Three points: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
7. Square, then subtract four.
a. Algebraic notation: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
b. Three points: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
8. Add three. Then, multiply by four. Then, subtract twelve. Then, divide by the original number.
a. Algebraic notation: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
b. Three points: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
9. In some of the above cases, two functions always give the same answer, even though they are different functions. We say that these functions are “equal” to each other. For instance, the function “add three and then subtract five” is equal to the function “subtract two” because they always give the same answer. (Try it, if you don’t believe me!) We can write this as:
\begin{align*}x+3-5=x-2\end{align*}
Note that this is not an equation you can solve for \begin{align*}x\end{align*}—it is a generalization which is true for all \begin{align*}x\end{align*} values. It is a way of indicating that if you do the calculation on the left, and the calculation on the right, they will always give you the same answer.
In the functions \begin{align*}2-8\end{align*} above, there are three such pairs of “equal” functions. Which ones are they? Write the algebraic equations that state their equalities (like my \begin{align*}x+3-5=x-2\end{align*} equation).
10. Of the following sets of numbers, there is one that could not possibly have been generated by any function whatsoever. Which set it is, and why? (No credit unless you explain why!)
a. \begin{align*}(3,6)(4,8)(-2,-4)\end{align*}
b. \begin{align*}(6,9)(2,9)(-3,9)\end{align*}
c. \begin{align*}(1,112)(2,-4)(3,3)\end{align*}
d. \begin{align*}(3,4)(3,9)(4,10)\end{align*}
e. \begin{align*}(-2,4)(-1,1)(0,0)(1,1)(2,4)\end{align*}
Name: \begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
Homework: Functions in the Real World
Laura is selling doughnuts for \begin{align*}35{\cancel{\mathrm{c}}}\end{align*} each. Each customer fills a box with however many doughnuts he wants, and then brings the box to Laura to pay for them. Let \begin{align*}n\end{align*} represent the number of doughnuts in a box, and let \begin{align*}c\end{align*} represent the cost of the box (in cents).
a. If the box has \begin{align*}3\end{align*} doughnuts, how much does the box cost?
b. If \begin{align*}c=245\end{align*}, how much does the box cost? How many doughnuts does it have?
c. If a box has \begin{align*}n\end{align*} doughnuts, how much does it cost?
d. Write a function \begin{align*}c(n)\end{align*} that gives the cost of a box, as a function of the number of doughnuts in the box.
2. Worth is doing a scientific study of graffiti in the downstairs boy’s room. On the first day of school, there is no graffiti. On the second day, there are two drawings. On the third day, there are four drawings. He forgets to check on the fourth day, but on the fifth day, there are eight drawings. Let \begin{align*}d\end{align*} represent the day, and \begin{align*}g\end{align*} represent the number of graffiti marks that day.
a. Fill in the following table, showing Worth’s four data points.
\begin{align*}& \text{d (day)} \\ & \text{g (number of graffiti marks)}\end{align*}
b. If this pattern keeps up, how many graffiti marks will there be on day \begin{align*}10?\end{align*}
c. If this pattern keeps up, on what day will there be \begin{align*}40\end{align*} graffiti marks?
d. Write a function \begin{align*}g(d)\end{align*} that gives the number of graffiti marks as a function of the day.
3. Each of the following is a set of points. Next to each one, write “yes” if that set of points could have been generated by a function, and “no” if it could not have been generated by a function. (You do not have to figure out what the function is. But you may want to try for fun—I didn’t just make up numbers randomly...)
a. \begin{align*}(1,-1)(3,-3)(-1,-1)(-3,-3) \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
b. \begin{align*}(1,\pi)(3,\pi)(9,\pi)(\pi,\pi) \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
c. \begin{align*}(1,1)(-1,1)(2,4)(-2,4)(3,9)(-3,9) \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
d. \begin{align*}(1,1)(1,-1)(4,2)(4,-2)(9,3)(9,-3) \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
e. \begin{align*}(1,1)(2,3)(3,6)(4,10) \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}f(x)=x^2+2x+1.\end{align*}
a. \begin{align*}f(2)=\end{align*}
b. \begin{align*}f(-1)=\end{align*}
c. \begin{align*}f(\frac{3}{2})=\end{align*}
d. \begin{align*}f(y)=\end{align*}
e. \begin{align*}f(\mathrm{spaghetti})=\end{align*}
f. \begin{align*}f(\sqrt{x})=\end{align*}
g. \begin{align*}f(f(x))=\end{align*}
5. Make up a function that has something to do with movies.
a. Think of a scenario where there are two numbers, one of which depends on the other. Describe the scenario, clearly identifying the independent variable and the dependent variable.
b. Write the function that shows how the dependent variable depends on the independent variable.
c. Now, plug in an example number to show how it works.
Algebraic Generalizations
1. a. Pick a number: _____
b. Add three: _____
c. Subtract three from your answer in part (b): _____
d. What happened? ______________________________________________
e. Write an algebraic generalization to represent this rule. _________________
f. Is there any number for which this rule will not work? ___________________
2. a. Pick a number: _____
b. Subtract five: _____
c. Double your answer in part (b): _____
d. Add ten to your answer in part (c): _____
e. Divide your answer in part (d) by your original number (a): _____
f. Now, repeat that process for three different numbers. Record the number you started with (a) and the number you ended up with (e).
\begin{align*}& \text{Started \ with:} \underline{\;\;\;\;\;\;\;\;\;} && \text{Started \ with:} \underline{\;\;\;\;\;\;\;\;\;} && \text{Started \ with:} \underline{\;\;\;\;\;\;\;\;\;} \\ & \text{Ended \ with:} \underline{\;\;\;\;\;\;\;\;\;} && \text{Ended \ with:} \underline{\;\;\;\;\;\;\;\;\;} && \text{Ended \ with:} \underline{\;\;\;\;\;\;\;\;\;} \end{align*}
g. What happened? ______________________________________________
h. Write an algebraic generalization to represent this rule. _________________
i. Is there any number for which this rule will not work? ___________________
3. Here are the first six powers of two.
- \begin{align*}2^1 = 2\end{align*}
- \begin{align*}2^2 = 4\end{align*}
- \begin{align*}2^3 = 8\end{align*}
- \begin{align*}2^4 = 16\end{align*}
- \begin{align*}2^5 = 32\end{align*}
- \begin{align*}2^6 = 64\end{align*}
a. If I asked you for \begin{align*}2^7\end{align*} (without a calculator), how would you get it? More generally, how do you always get from one term in this list to the next term? ________________
b. Write an algebraic generalization to represent this rule. _____________________
4. Now, we’re going to make that rule even more general. Suppose I want to multiply \begin{align*}2^5\end{align*} times \begin{align*}2^3\end{align*}. Well, \begin{align*}2^5\end{align*} means \begin{align*}2^*2^*2^*2^*2\end{align*}, and \begin{align*}2^3\end{align*} means \begin{align*}2^*2^*2\end{align*}. So we can write the whole thing out like this.
\begin{align*}& \qquad \quad 2^5 \qquad \qquad \qquad \quad 2^3 \qquad = \qquad \qquad \qquad 2^8 \\ & \overbrace{2*2*2*2*2} \quad * \quad \overbrace{2*2*2} \quad = \quad \overbrace{2*2*2*2*2*2*2*2}\end{align*}
This shows that (\begin{align*}2^5\end{align*})(\begin{align*}2^3\end{align*}) = \begin{align*}2^8\end{align*}.
a. Using a similar drawing, demonstrate what (\begin{align*}10^3\end{align*})(\begin{align*}10^4\end{align*}) must be.
b. Now, write an algebraic generalization for this rule. ____________________
5. The following statements are true.
- \begin{align*}3 \times 4 = 4 \times 3\end{align*}
- \begin{align*}7 \times -3 = -3 \times 7\end{align*}
- \begin{align*}\frac{1}{2} \times 8 = 8 \times \frac{1}{2}\end{align*}
Write an algebraic generalization for this rule. __________________
6. Look at the following pairs of statements.
\begin{align*}& 8 \times 8 = 64 && 5 \times 5 = 25 && 10 \times 10 = 100 && 3 \times 3 = 9 \\ & 7 \times 9 = 63 && 4 \times 6 = 24 && 9 \times 11 = 99 && 2 \times 4 = 8\end{align*}
a. Based on these pairs, if I told you that \begin{align*}30 \times 30=900\end{align*}, could you tell me (immediately, without a calculator) what \begin{align*}29 \times 31\end{align*} is? ____________________________
b. Express this rule—the pattern in these numbers—in words.
c. Whew! That was complicated, wasn’t it? Good thing we have math. Write the algebraic generalization for this rule. _______________________
d. Try out this generalization with negative numbers, with zero, and with fractions. (Show your work below, trying all three of these cases separately.) Does it always work, or are there cases where it doesn’t?
Name: __________________
Homework: Algebraic Generalizations
In class, we talked about the following four pairs of statements.
\begin{align*}& 8 \times 8 = 64 && 5 \times 5 = 25 && 10 \times 10 = 100 && 3 \times 3 = 9 \\ & 7 \times 9 = 63 && 4 \times 6 = 24 && 9 \times 11 = 99 && 2 \times 4 = 8\end{align*}
You made an algebraic generalization about these statements: write that generalization again below.
Now, we are going to generalize it further. Let’s focus on the \begin{align*}10 \times 10\end{align*} thing.
\begin{align*}10 \times 10 = 100\end{align*}
There are two numbers that are one away from \begin{align*}10\end{align*}; these numbers are, of course, \begin{align*}9\end{align*} and \begin{align*}11\end{align*}. As we saw, \begin{align*}9 \times 11\end{align*} is \begin{align*}99\end{align*}. It is one less than \begin{align*}100\end{align*}.
Now, suppose we look at the two numbers that are two away from \begin{align*}10?\end{align*} Or three away? Or four away? We get a sequence like this (fill in all the missing numbers):
\begin{align*}10 \times 10 & = 100\\ 9 \times 11 &= 99 && 1 \ \text{away \ from} \ 10, \ \text{the \ product \ is} \ 1 \ \text{less \ than} \ 100 \\ 8 \times 12 &= \underline{\;\;\;\;\;\;} && 2 \ \text{away \ from} \ 10, \ \text{the \ product \ is} \ \underline{\;\;\;\;\;\;} \text{less \ than} \ 100 \\ 7 \times 13 &= \underline{\;\;\;\;\;\;} && 3 \ \text{away \ from} \ 10, \ \text{the \ product \ is} \ \underline{\;\;\;\;\;\;} \text{less \ than} \ 100 \\ \underline{\;\;\;\;} \times \underline{\;\;\;\;} &= \underline{\;\;\;\;\;\;} && \underline{\;\;\;} \text{away \ from} \ 10, \ \text{the \ product \ is} \underline{\;\;\;\;\;\;} \text{less \ than} \ 100 \\ \underline{\;\;\;\;} \times \underline{\;\;\;\;} &= \underline{\;\;\;\;\;\;} && \underline{\;\;\;} \text{away \ from} \ 10, \ \text{the \ product \ is} \underline{\;\;\;\;\;\;} \text{less \ than} \ 100\end{align*}
Do you see the pattern? What would you expect to be the next sentence in this sequence?
Write the algebraic generalization for this rule.
Does that generalization work when the “___away from \begin{align*}10\end{align*}” is \begin{align*}0?\end{align*} Is a fraction? Is a negative number? Test all three cases. (Show your work!)
Name: __________________
Homework: Graphing
The following graph shows the temperature throughout the month of March. Actually, I just made this graph up—the numbers do not actually reflect the temperature throughout the month of March. We’re just pretending, OK?
1. Give a weather report for the month of March, in words.
2. On what days was the temperature exactly \begin{align*}0^{o}C?\end{align*}
3. On what days was the temperature below freezing?
4. On what days was the temperature above freezing?
5. What is the domain of this graph?
6. During what time periods was the temperature going up?
7. During what time periods was the temperature going down?
8. Mary started a company selling French Fries over the Internet. For the first \begin{align*}3\end{align*} days, while she worked on the technology, she lost \begin{align*}\$100\end{align*} per day. Then she opened for business. People went wild over her French Fries! She made \begin{align*}\$200\end{align*} in one day, \begin{align*}\$300\end{align*} the day after that, and \begin{align*}\$400\end{align*} the day after that. The following day she was sued by an angry customer who discovered that Mary had been using genetically engineered potatoes. She lost \begin{align*}\$500\end{align*} in the lawsuit that day and closed up her business. Draw a graph showing Mary’s profits as a function of days.
9. Fill in the following table. Then, based on your table, draw graphs of the functions \begin{align*}y=x^2, y=x^2+2, y=x^2-1, y=(x+3)^2, y=2x^2,\end{align*} and \begin{align*}y=-x^2\end{align*}.
\begin{align*}x\end{align*} | \begin{align*}x^2\end{align*} | \begin{align*}x^2+2\end{align*} | \begin{align*}x^2-1\end{align*} | \begin{align*}(x+3)^2\end{align*} | \begin{align*}2x^2\end{align*} | \begin{align*}-x^2\end{align*} |
---|---|---|---|---|---|---|
\begin{align*}-3\end{align*} | ||||||
\begin{align*}-2\end{align*} | ||||||
\begin{align*}-1\end{align*} | ||||||
\begin{align*}0\end{align*} | ||||||
\begin{align*}1\end{align*} | ||||||
\begin{align*}2\end{align*} | ||||||
\begin{align*}3\end{align*} |
Now describe in words what happened...
a. How did adding \begin{align*}2\end{align*} to the function change the graph?
b. How did subtracting \begin{align*}1\end{align*} from the function change the graph?
c. How did adding three to \begin{align*}x\end{align*} change the graph?
d. How did doubling the function change the graph?
e. How did multiplying the graph by \begin{align*}-1\end{align*} change the graph?
f. By looking at your graphs, estimate the point of intersection of the graphs \begin{align*}y=x^2\end{align*} and \begin{align*}y=(x+3)^2\end{align*}. What does this point represent?
Name: __________________
Horizontal and Vertical Permutations
1.Standing at the edge of the Bottomless Pit of Despair, you kick a rock off the ledge and it falls into the pit. The height of the rock is given by the function \begin{align*}h(t) = -16t^2\end{align*}, where \begin{align*}t\end{align*} is the time since you dropped the rock, and \begin{align*}h\end{align*} is the height of the rock.
a. Fill in the following table.
\begin{align*}&\text{time \ (seconds)} && 0 && \frac{1}{2} && 1 && 1 \frac{1}{2} && 2 && 2 \frac{1}{2} && 3 && 3 \frac{1}{2}\\ &\text{height \ (feet)}\end{align*}
b. \begin{align*}h(0)=0\end{align*}. What does that tell us about the rock?
c. All the other heights are negative: what does that tell us about the rock?
d. Graph the function \begin{align*}h(t)\end{align*}. Be sure to carefully label your axes!
2. Another rock was dropped at the exact same time as the first rock; but instead of being kicked from the ground, it was dropped from your hand, \begin{align*}3\;\mathrm{feet}\end{align*} up. So, as they fall, the second rock is always three feet higher than the first rock.
a. Fill in the following table for the second rock.
\begin{align*}&\text{time \ (seconds)} && 0 && \frac{1}{2} && 1 && 1\frac{1}{2} && 2 && 2\frac{1}{2} && 3 && 3\frac{1}{2}\\ & \text{height \ (feet)}\end{align*}
b. Graph the function \begin{align*}h(t)\end{align*} for the new rock. Be sure to carefully label your axes!
c. How does this new function \begin{align*}h(t)\end{align*} compare to the old one? That is, if you put them side by side, what change would you see?
d. The original function was \begin{align*}h(t) = -16t^2\end{align*}. What is the new function? \begin{align*}h(t)=\end{align*}
(*make sure the function you write actually generates the points in your table!)
e. Does this represent a horizontal permutation or a vertical permutation?
f. Write a generalization based on this example, of the form: when you do such-and-such to a function, the graph changes in such-and-such a way.
3. A third rock was dropped from the exact same place as the first rock (kicked off the ledge), but it was dropped \begin{align*}1\frac{1}{2}\end{align*} seconds later, and began its fall (at \begin{align*}h=0\end{align*}) at that time.
a. Fill in the following table for the third rock.
\begin{align*}&\text{time \ (seconds)} && 0 && \frac{1}{2} && 1 && 1\frac{1}{2} && 2 && 2\frac{1}{2} && 3 && 3\frac{1}{2} && 4 && 4\frac{1}{2} && 5\\ &\text{height \ (feet)} && 0 && 0 && 0 && 0\end{align*}
b. Graph the function \begin{align*}h(t)\end{align*} for the new rock. Be sure to carefully label your axes!
c. How does this new function \begin{align*}h(t)\end{align*} compare to the original one? That is, if you put them side by side, what change would you see?
d. The original function was \begin{align*}h(t) = -16t^2\end{align*}. What is the new function? \begin{align*}h(t)=\end{align*}
(*make sure the function you write actually generates the points in your table!)
e. Does this represent a horizontal permutation or a vertical permutation?
f. Write a generalization based on this example, of the form: when you do such-and-such to a function, the graph changes in such-and-such a way.
Name: __________________
Homework: Horizontal and Vertical Permutations
1. In a certain magical bank, your money doubles every year. So if you start with \begin{align*}^{\$}1\end{align*}, your money is represented by the function \begin{align*}M=2^t\end{align*}, where \begin{align*}t\end{align*} is the time (in years) your money has been in the bank, and \begin{align*}M\end{align*} is the amount of money (in dollars) you have.
Don puts \begin{align*}^{\$}1\end{align*} into the bank at the very beginning (\begin{align*}t=0\end{align*}).
Susan also puts \begin{align*}^{\$}1\end{align*} into the bank when \begin{align*}t=0\end{align*}. However, she also has a secret stash of \begin{align*}^{\$}2\end{align*} under her mattress at home. Of course, her \begin{align*}^{\$}2\end{align*} stash doesn’t grow: so at any given time \begin{align*}t\end{align*}, she has the same amount of money that Don has, plus \begin{align*}^{\$}2\end{align*} more.
Cheryl, like Don, starts with \begin{align*}^{\$}1\end{align*}. But during the first year, she hides it under her mattress. After a year (\begin{align*}t=1\end{align*}) she puts it into the bank, where it starts to accrue interest.
a. Fill in the following table to show how much money each person has.
\begin{align*}t=0\end{align*} | \begin{align*}t=1\end{align*} | \begin{align*}t=2\end{align*} | \begin{align*}t=3\end{align*} | |
---|---|---|---|---|
Don | \begin{align*}1\end{align*} | |||
Susan | \begin{align*}3\end{align*} | |||
Cheryl | \begin{align*}1\end{align*} | \begin{align*}1\end{align*} |
b. Graph each person’s money as a function of time.
c. Below each graph, write the function that gives this person’s money as a function of time. Be sure your function correctly generates the points you gave above! (*For Cheryl, your function will not accurately represent her money between \begin{align*}t=0\end{align*} and \begin{align*}t=1\end{align*}, but it should accurately represent it thereafter.)
2. The function \begin{align*}y=f(x)\end{align*} is defined on the domain \begin{align*}[-4,4]\end{align*} as shown below.
a. What is \begin{align*}f(-2)?\end{align*} (That is, what does this function give you if you give it a -\begin{align*}2?\end{align*})
b. What is \begin{align*}f(0)?\end{align*}
c. What is \begin{align*}f(3)?\end{align*}
d. The function has three zeros. What are they?
The function \begin{align*}g(x)\end{align*} is defined by the equation: \begin{align*}g(x)=f(x)-1\end{align*}. That is to say, for any \begin{align*}x\end{align*}-value you put into \begin{align*}g(x)\end{align*}, it first puts that value into \begin{align*}f(x)\end{align*}, and then it subtracts \begin{align*}1\end{align*} from the answer.
e. What is \begin{align*}g(-2)?\end{align*}
f. What is \begin{align*}g(0)?\end{align*}
g. What is \begin{align*}g(3)?\end{align*}
h. Draw \begin{align*}y=g(x)\end{align*} next to the \begin{align*}f(x)\end{align*} drawing above.
The function \begin{align*}h(x)\end{align*} is defined by the equation: \begin{align*}h(x)=f(x+1)\end{align*}. That is to say, for any \begin{align*}x-\end{align*}value you put into \begin{align*}h(x)\end{align*}, it first adds \begin{align*}1\end{align*} to that value, and then it puts the new \begin{align*}x-\end{align*}value into \begin{align*}f(x)\end{align*}.
i. What is \begin{align*}h(-3)?\end{align*}
j. What is \begin{align*}h(-1)?\end{align*}
k. What is \begin{align*}h(2)?\end{align*}
l. Draw \begin{align*}y=h(x)\end{align*} next to the \begin{align*}f(x)\end{align*} drawing to the right.
m. Which of the two permutations above changed the domain of the function?
3. On your calculator, graph the function \begin{align*}Y1=x^3-13x-12\end{align*}. Graph it in a window with \begin{align*}x\end{align*} going from \begin{align*}-5\end{align*} to \begin{align*}5\end{align*}, and \begin{align*}y\end{align*} going from \begin{align*}-30\end{align*} to \begin{align*}30\end{align*}.
a. Copy the graph below. Note the three zeros at \begin{align*}x=-3, x=-1,\end{align*} and \begin{align*} x=4\end{align*}.
b. For what \begin{align*}x-\end{align*}values is the function less than zero? (Or, to put it another way: solve the inequality \begin{align*}x^3-13x-12 < 0\end{align*}.)
c. Construct a function that looks exactly like this function, but moved up \begin{align*}10\end{align*}. Graph your new function on the calculator (as \begin{align*}Y2\end{align*}, so you can see the two functions together). When you have a function that works, write your new function below.
d. Construct a function that looks exactly like the original function, but moved \begin{align*}2\;\mathrm{units}\end{align*} to the left. When you have a function that works, write your new function below.
e. Construct a function that looks exactly like the original function, but moved down \begin{align*}3\end{align*} and \begin{align*}1\;\mathrm{unit}\end{align*} to the right. When you have a function that works, write your new function below.
Name: __________________
Sample Test: Functions I
1. Chris is \begin{align*}1\frac{1}{2}\end{align*} years younger than his brother David. Let \begin{align*}D\end{align*} represent David’s age, and \begin{align*}C\end{align*} represent Chris’s age.
a. If Chris is fifteen years old, how old is David? ______
b. Write a function to show how to find David’s age, given Chris’s age. \begin{align*} D(C)=\end{align*}_____
2. Sally slips into a broom closet, waves her magic wand, and emerges as...the candy bar fairy! Flying through the window of the classroom, she gives every student two candy bars. Then five candy bars float through the air and land on the teacher’s desk. And, as quickly as she appeared, Sally is gone to do more good in the world.
Let \begin{align*}s\end{align*} represent the number of students in the class, and \begin{align*}c\end{align*} represent the total number of candy bars distributed. Two for each student, and five for the teacher.
a. Write a function to show how many candy bars Sally gave out, as a function of the number of students. \begin{align*}c(s)=\end{align*}_____
b. Use that function to answer the question: if there were \begin{align*}20\end{align*} students in the classroom, how many candy bars were distributed? First represent the question in functional notation—then answer it. __________
c. Now use the same function to answer the question: if Sally distributed \begin{align*}35\end{align*} candy bars, how many students were in the class? First represent the question in functional notation—then answer it. __________
3. The function \begin{align*}f(x)\end{align*} is “Subtract three, then take the square root.”
a. Express this function algebraically, instead of in words: \begin{align*}f(x) =\end{align*} ________________
b. Give any three points that could be generated by this function: _____________________
c. What \begin{align*}x-\end{align*}values are in the domain of this function? _____________________
4. The function \begin{align*}y(x)\end{align*} is “Given any number, return \begin{align*}6\end{align*}.”
a. Express this function algebraically, instead of in words: \begin{align*}y(x) =\end{align*} _______________
b. Give any three points that could be generated by this function: _____________________
c. What \begin{align*}x-\end{align*}values are in the domain of this function? _____________________
5. \begin{align*}z(x)=x^2-6x+9\end{align*}
a. \begin{align*}z(-1) = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
b. \begin{align*}z(0) = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
c. \begin{align*}z(1) = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
d. \begin{align*}z(3) = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
e. \begin{align*}z(x+2) = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
f. \begin{align*}z(z(x)) = \underline{\;\;\;\;\;\;\;\;\;}\end{align*}
6. Of the following sets of numbers, indicate which ones could possibly have been generated by a function. All I need is a “Yes” or “No”—you don’t have to tell me the function! (But, you may do so if you would like to...)
a. \begin{align*}(-2,4)(-1,1)(0,0)(1,1)(2,4) \qquad \ \Box \ \text{Yes} \qquad \Box \ \text{No}\end{align*}
b. \begin{align*}(4,-2)(1,-1)(0,0)(1,1)(4,2) \ \ \quad \Box \ \text{Yes} \qquad \Box \ \text{No}\end{align*}
c. \begin{align*}(2,\pi)(3,\pi)(4,\pi)(5,1) \qquad \qquad \quad \Box \ \text{Yes} \qquad \Box \ \text{No}\end{align*}
d. \begin{align*}(\pi,2)(\pi,3)(\pi,4)(1,5) \qquad \qquad \quad \Box \ \text{Yes} \qquad \Box \ \text{No}\end{align*}
7. Make up a function involving music.
a. Write the scenario. Your description should clearly tell me—in words—how one value depends on another.
b. Name, and clearly describe, two variables. Indicate which is dependent and which is independent.
c. Write a function showing how the dependent variable depends on the independent variable. If you were explicit enough in parts (a) and (b), I should be able to predict your answer to part (c) before I read it.
d. Choose a sample number to show how your function works. Explain what the result means.
8. Here is an algebraic generalization: for any number \begin{align*}x, \ x^2-25=(x+5)(x-5)\end{align*}.
a. Plug \begin{align*}x=3\end{align*} into that generalization, and see if it works.
b. \begin{align*}20 \times 20\end{align*} is \begin{align*}400\end{align*}. Given that, and the generalization, can you find \begin{align*}15 \times 25\end{align*} without a calculator? (Don’t just give me the answer, show how you got it!)
9. Amy has started a company selling candy bars. Each day, she buys candy bars from the corner store and sells them to students during lunch. The following graph shows her profit each day in March.
a. On what days did she break even?
b. On what days did she lose money?
10. The picture to the right shows the graph of \begin{align*} y=\sqrt{x}\end{align*}. The graph starts at \begin{align*}(0,0)\end{align*} and moves up and to the right forever.
a. What is the domain of this graph?
b. Write a function that looks exactly the same, except that it starts at the point \begin{align*}(-3,1)\end{align*} and moves up-and-right from there.
11. The following graph represents the graph \begin{align*}y=f(x)\end{align*}.
a. Is it a function? Why or why not?
b. What are the zeros?
c. For what \begin{align*}x-\end{align*}values is it positive?
d. For what \begin{align*}x-\end{align*}values is it negative?
e. To the right is the same function \begin{align*}f(x)\end{align*}. On that same graph, draw the graph of \begin{align*}y = f(x)-2\end{align*}.
f. To the right is the same function \begin{align*} f(x)\end{align*}. On that same graph, draw the graph of \begin{align*}y = -f(x)\end{align*}.
Extra credit: Here is a cool trick for squaring a difficult number, if the number immediately below it is easy to square.
Suppose I want to find \begin{align*}31^2\end{align*}. That’s hard. But it’s easy to find \begin{align*}30^2\end{align*}, that’s \begin{align*}900\end{align*}. Now, here comes the trick: add \begin{align*}30\end{align*}, and then add \begin{align*}31\end{align*}. \begin{align*}900+30+31=961\end{align*}. That’s the answer! \begin{align*}31^2 = 961\end{align*}.
a. Use this trick to find \begin{align*}41^2\end{align*}. (Don’t just show me the answer, show me the work!)
b. Write the algebraic generalization that represents this trick.
Name: __________________
Lines
1. You have \begin{align*}\$150\end{align*} at the beginning of the year. (Call that day “\begin{align*}0\end{align*}.”) Every day you make \begin{align*}\$3\end{align*}.
a. How much money do you have on day \begin{align*}1?\end{align*}
b. How much money do you have on day \begin{align*}4?\end{align*}
c. How much money do you have on day \begin{align*}10?\end{align*}
d. How much money do you have on day \begin{align*}n?\end{align*} This gives you a general function for how much money you have on any given day.
e. How much is that function going up every day? This is the slope of the line.
f. Graph the line.
2. Your parachute opens when you are \begin{align*}2,000 \;\mathrm{feet}\end{align*} above the ground. (Call this time \begin{align*}t=0\end{align*}.) Thereafter, you fall \begin{align*}30 \;\mathrm{feet}\end{align*} every second. (Note: I don’t know anything about skydiving, so these numbers are probably not realistic!)
a. How high are you after one second?
b. How high are you after ten seconds?
c. How high are you after fifty seconds?
d. How high are you after \begin{align*}t\end{align*} seconds? This gives you a general formula for your height.
e. How long does it take you to hit the ground?
f. How much altitude are you gaining every second? This is the slope of the line. Because you are falling, you are actually gaining negative altitude, so the slope is negative.
g. Graph the line.
3. Make up a word problem like numbers \begin{align*}1\end{align*} and \begin{align*}2\end{align*}. Be very clear about the independent and dependent variables, as always. Make sure the relationship between them is linear! Give the general equation and the slope of the line.
4. Compute the slope of a line that goes from \begin{align*}(1,3)\end{align*} to \begin{align*}(6,18)\end{align*}.
5. For each of the following diagrams, indicate roughly what the slope is.
a.
b.
c.
d.
e.
f.
6. Now, for each of the following graphs, draw a line with roughly the slope indicated. For instance, on the first little graph, draw a line with slope \begin{align*}2\end{align*}.
a.
b.
c.
For problems \begin{align*}7\end{align*} and \begin{align*}8\end{align*},
- Solve for \begin{align*}y\end{align*}, and put the equation in the form \begin{align*}y=mx+b\end{align*} (...if it isn’t already in that form)
- Identify the slope
- Identify the \begin{align*}y-\end{align*} intercept, and graph it
- Use the slope to find one point other than the \begin{align*}y-\end{align*} intercept on the line
- Graph the line
7. \begin{align*}y=3x-2\end{align*}
Slope: ___________
\begin{align*}y-\end{align*} intercept: ___________
Other point: ___________
8. \begin{align*}2y-x=4\end{align*}
Equation in \begin{align*}y=mx+b\end{align*} form:
Slope: ___________
\begin{align*}y-\end{align*} intercept: ___________
Other point: ___________
Name: ___________________________
Homework: Graphing Lines
1. \begin{align*}2y+7x+3=0\end{align*} is the equation for a line.
a. Put this equation into the “slope-intercept” form \begin{align*}y=mx+b\end{align*}
b. slope = ___________
c. \begin{align*} y-\end{align*} intercept = ___________
d. \begin{align*}x-\end{align*} intercept = ___________
e. Graph it
2. The points \begin{align*}(5,2)\end{align*} and \begin{align*}(7,8)\end{align*} lie on a line.
a. Find the slope of this line
b. Find another point on this line
3. When you’re building a roof, you often talk about the “pitch” of the roof—which is a fancy word that means its slope. You are building a roof shaped like the following. The roof is perfectly symmetrical. The slope of the left-hand side is \begin{align*}\frac{1}{3}\end{align*}. In the drawing below, the roof is the two thick black lines—the ceiling of the house is the dotted line \begin{align*}60'\end{align*} long.
a. What is the slope of the right-hand side of the roof?
b. How high is the roof? That is, what is the distance from the ceiling of the house, straight up to the point at the top of the roof?
c. How long is the roof? That is, what is the combined length of the two thick black lines in the drawing above?
4. In the equation \begin{align*}y=3x\end{align*}, explain why \begin{align*}3\end{align*} is the slope. (Don’t just say, “because it’s the \begin{align*}m\end{align*} in \begin{align*}y=mx+b\end{align*}.” Explain why \begin{align*}\frac{\triangle {y}}{\triangle {x}}\end{align*} will be \begin{align*}3\end{align*} for any two points on this line, just like we explained in class why \begin{align*}b\end{align*} is the \begin{align*}y-\end{align*}intercept.)
5. How do you measure the height of a very tall mountain? You can’t just sink a ruler down from the top to the bottom of the mountain!.
So here’s one way you could do it. You stand behind a tree, and you move back until you can look straight over the top of the tree to the top of the mountain. Then you measure the height of the tree, the distance from you to the mountain, and the distance from you to the tree. So you might get results like this.
How high is the mountain?
6. The following table shows how much money Scrooge McDuck has been worth every year since 1999.
\begin{align*}&\text{Year} && 1999 && 2000 && 2001 && 2002 && 2003 && 2004\\ &\text{Net \ worth} && \$ 3 \ \text{Trillion} && \$ 4.5 \ \text{Trillion} && \$ 6 \ \text{Trillion} && \$ 7.5 \ \text{Trillion} && \$ 9 \ \text{Trillion} && \$ 10.5 \ \text{Trillion}\end{align*}
a. How much is a trillion, anyway?
b. Graph this relation.
c. What is the slope of the graph?
d. How much money can Mr. McDuck earn in \begin{align*}20\end{align*} years at this rate?
7. Make up and solve your own word problem using slope.
Composite Functions
1. You are the foreman at the Sesame Street Number Factory. A huge conveyer belt rolls along, covered with big plastic numbers for our customers. Your two best employees are Katie and Nicolas. Both of them stand at their stations by the conveyer belt. Nicolas’s job is: whatever number comes to your station, add \begin{align*}2\end{align*} and then multiply by \begin{align*}5\end{align*}, and send out the resulting number. Katie is next on the line. Her job is: whatever number comes to you, subtract \begin{align*}10\end{align*}, and send the result down the line to Sesame Street.
a. Fill in the following table.
\begin{align*}&\text{This number comes the line} && -5 && -3 && -1 && 2 && 4 && 6 && 10 && x && 2x\\ &\text{Nicolas comes up with this number, and} \\ &\text{sends it down the line to Katie}\\ &\text{Katie\ then spits out this number}\end{align*} b. In a massive downsizing effort, you are going to fire Nicolas. Katie is going to take over both functions (Nicolas’s and her own). Katie is given a number; first she takes care of Nicolas's function, and then her own. But now Katie is overworked, so she comes up with a shortcut: one function she can do, that covers both Nicolas’s job and her own. What does Katie do to each number you give her? (Answer in words.)
2. Taylor is driving a motorcycle across the country. Each day he covers \begin{align*}500 \;\mathrm{miles}\end{align*}. A policeman started the same place Taylor did, waited a while, and then took off, hoping to catch some illegal activity. The policeman stops each day exactly five miles behind Taylor.
Let \begin{align*}d\end{align*} equal the number of days they have been driving. (So after the first day, \begin{align*}d=1\end{align*}.) Let \begin{align*}T\end{align*} be the number of miles Taylor has driven. Let \begin{align*}p\end{align*} equal the number of miles the policeman has driven.
a. After three days, how far has Taylor gone? ______________
b. How far has the policeman gone? ______________
c. Write a function \begin{align*}T(d)\end{align*} that gives the number of miles Taylor has traveled, as a function of how many days he has been traveling. ______________
d. Write a function \begin{align*}p(T)\end{align*} that gives the number of miles the policeman has traveled, as a function of the distance that Taylor has traveled. ______________
e. Now write the composite function \begin{align*} p(T(d))\end{align*} that gives the number of miles the policeman has traveled, as a function of the number of days he has been traveling.
3. Rashmi is an honor student by day; but by night, she works for the university as a tutor. Each month she gets paid a \begin{align*}\$1000\end{align*} base salary, plus an extra \begin{align*}\$100\end{align*} for each person she tutors. The university pays all tutors in \begin{align*}\$20\end{align*} bills.
Let \begin{align*}k\end{align*} equal the number of people Rashmi kills tutors in a given month. Let \begin{align*}m\end{align*} be the amount of money she is paid that month, in dollars. Let \begin{align*}b\end{align*} be the number of \begin{align*}\$20\end{align*} bills she gets.
a. Write a function \begin{align*}m(k)\end{align*} that tells how much money Rashmi makes, in a given month, as a function of the number of people she tutors. ______________
b. Write a function \begin{align*}b(m)\end{align*} that tells how many bills Rashmi gets, in a given month, as a function of the number of dollars she makes. ______________
c. Write a composite function \begin{align*}b(m(k))\end{align*} that gives the number of bills Rashmi gets, as a function of the number of people she tutors.
d. If Rashmi kills \begin{align*}5\end{align*} students in a month, how many \begin{align*}\$20\end{align*} bills does she earn? First, translate this question into function notation—then solve it for a number.
e. If Rashmi earns \begin{align*}100 \ \$20\end{align*} bills in a month, how many students did she tutor? First, translate this question into function notation—then solve it for a number.
4. Make up a problem like numbers \begin{align*}2\end{align*} and \begin{align*}3\end{align*}. Be sure to take all the right steps: define the scenario, define your variables clearly, and then show the functions that relate the variables. This is just like the problems we did last week, except that you have to use three variables, related by a composite function.
5. \begin{align*}f(x)=\sqrt{x=2}. \ g(x)=x^2+x.\end{align*}
a. \begin{align*}f(7) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
b. \begin{align*}g(7) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
c. \begin{align*}f(g(x)) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
d. \begin{align*}f(f(x)) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
e. \begin{align*}g(f(x)) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
f. \begin{align*}g(g(x)) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
g. \begin{align*}f(g(3)) = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}h(x)=x-5. \ h(i(x))=x\end{align*}. Can you find what function \begin{align*}i(x)\end{align*} is, to make this happen?
Name: __________________
Homework: Composite Functions
1. An inchworm (exactly one inch long, of course) is crawling up a yardstick (guess how long that is?). After the first day, the inchworm’s head (let’s just assume that’s at the front) is at the \begin{align*}3"\end{align*} mark. After the second day, the inchworm’s head is at the \begin{align*}6"\end{align*} mark. After the third day, the inchworm’s head is at the \begin{align*}9"\end{align*} mark.
Let \begin{align*}d\end{align*} equal the number of days the worm has been crawling. (So after the first day, \begin{align*}d=1\end{align*}.)
Let \begin{align*}h\end{align*} be the number of inches the head has gone. Let \begin{align*}t\end{align*} be the position of the worm’s tail.
a. After \begin{align*}10\end{align*} days, where is the inchworm’s head? ______________
b. Its tail? ______________
c. Write a function \begin{align*}h(d)\end{align*} that gives the number of inches the head has traveled, as a function of how many days the worm has been traveling. ______________
d. Write a function \begin{align*}t(h)\end{align*} that gives the position of the tail, as a function of the position of the head. ______________
e. Now write the composite function \begin{align*}t(h(d))\end{align*} that gives the position of the tail, as a function of the number of days the worm has been traveling.
2. The price of gas started out at \begin{align*}100\cancel{\mathrm{c}}/\mathrm{gallon}\end{align*} on the \begin{align*}1^\mathrm{st}\end{align*} of the month. Every day since then, it has gone up \begin{align*}2\cancel{\mathrm{c}}/\mathrm{gallon}\end{align*}. My car takes \begin{align*}10 \;\mathrm{gallons}\end{align*} of gas. (As you might have guessed, these numbers are all fictional.)
Let \begin{align*}d\end{align*} equal the date (so the \begin{align*}1^\mathrm{st}\end{align*} of the month is \begin{align*}1\end{align*}, and so on). Let \begin{align*}g\end{align*} equal the price of a gallon of gas, in cents. Let \begin{align*}c\end{align*} equal the total price required to fill up my car, in cents.
a. Write a function \begin{align*}g(d)\end{align*} that gives the price of gas on any given day of the month. ______________
b. Write a function \begin{align*}c(g)\end{align*} that tells how much money it takes to fill up my car, as a function of the price of a gallon of gas. ______________
c. Write a composite function \begin{align*}c(g(d))\end{align*} that gives the cost of filling up my car on any given day of the month.
d. How much money does it take to fill up my car on the 11th of the month? First, translate this question into function notation—then solve it for a number.
e. On what day does it cost \begin{align*}1,040\cancel{\mathrm{c}}\end{align*} (otherwise known as \begin{align*}\$10.40\end{align*}) to fill up my car? First, translate this question into function notation—then solve it for a number.
3. Make up a problem like numbers \begin{align*}1\end{align*} and \begin{align*}2\end{align*}. Be sure to take all the right steps: define the scenario, define your variables clearly, and then show the (composite) functions that relate the variables.
4. \begin{align*}f(x)=\frac{x}{x^2+3x+4}\end{align*}. Find \begin{align*}f(g(x))\end{align*} if...
a. \begin{align*}g(x)=3\end{align*}
b. \begin{align*}g(x)=y\end{align*}
c. \begin{align*}g(x)=\mathrm{oatmeal}\end{align*}
d. \begin{align*}g(x)=\sqrt{x}\end{align*}
e. \begin{align*}g(x)=(x+2)\end{align*}
f. \begin{align*}g(x)=\frac{x}{x^2+3x+4}\end{align*}
5. \begin{align*}h(x)=4x. \ h(i(x))=x.\end{align*} Can you find what function \begin{align*}i(x)\end{align*} is, to make this happen?
Name: ___________________________
Inverse Functions
1. We are playing the function game. Every time you give Christian a number, he doubles it and subtracts six.
a. If you give Christian a ten, what will he give you back?
b. If you give Christian an \begin{align*}x\end{align*}, what will he give you back?
c. What number would you give Christian, that would make him give you a \begin{align*}0?\end{align*}
d. What number would you give Christian, that would make him give you a ten?
e. What number would you give Christian, that would make him give you an \begin{align*}x?\end{align*} (Hint for the stuck: try to follow the process you used to answer part d.)
2. A television set dropped from the top of a \begin{align*}300'\end{align*} building falls according to the equation:
\begin{align*}h(t)=300-16t^2\end{align*}
where \begin{align*}t\end{align*} is the amount of time that has passed since it was dropped (measured in seconds), and \begin{align*}h\end{align*} is the height of the television set above ground (measured in feet).
a. Where is the television set after \begin{align*}0 \;\mathrm{seconds}\end{align*} have elapsed?
b. Where is the television set after \begin{align*}2 \;\mathrm{seconds}\end{align*} have elapsed?
c. A man is watching out of the window of the first floor, \begin{align*}20'\end{align*} above ground. At what time does the television set go flying by?
d. At what time does the television reach the ground?
e. Find a general formula \begin{align*}t(h)\end{align*} that can be used to quickly and easily answer all questions like (c) and (d).
Find the inverse of each function. For each one, check your answer by plugging in two different numbers to see if they work.
3. \begin{align*}y=x+5\end{align*}
- Inverse function:
- Test:
- Test:
4. \begin{align*}y=x-6\end{align*}
- Inverse function:
- Test:
- Test:
5. \begin{align*}y=3x\end{align*}
- Inverse function:
- Test:
- Test:
6. \begin{align*}y=\frac{x}{4}\end{align*}
- Inverse function:
- Test:
- Test:
7. \begin{align*}y=3x+12\end{align*}
- Inverse function:
- Test:
- Test:
8. \begin{align*}y=\frac{100}{x}\end{align*}
- Inverse function:
- Test:
- Test:
9. \begin{align*}y=\frac{2x+3}{7}\end{align*}
- Inverse function:
- Test:
- Test:
10. \begin{align*}y=x^2\end{align*}
- Inverse function:
- Test:
- Test:
11. \begin{align*}y=2^x\end{align*}
- Inverse function:
- Test:
- Test:
Name: __________________
Homework: Inverse Functions
1. On our last “Sample Test,” we did a scenario where Sally distributed two candy bars to each student and five to the teacher. We found a function \begin{align*}c(s)\end{align*} that represented how many candy bars she distributed, as a function of the number of students in the room.
a. What was that function again?
b. How many candy bars would Sally distribute if there were \begin{align*}20\end{align*} students in the room?
c. Find the inverse function.
d. Now—this is the key part—explain what that inverse function actually represents. Ask a word-problem question that I can answer by using the inverse function.
2. Make up a problem like \begin{align*}^\#1\end{align*}. That is, make up a scenario, and show the function that represents that scenario. Then, give a word problem that is answered by the inverse function, and show the inverse function.
For each function, find the inverse function, the domain, and the range.
3. \begin{align*}y=2+\frac{1}{x}\end{align*}
4. \begin{align*}\frac{2x+3}{7}\end{align*}
5. \begin{align*}2(x+3)\end{align*}
6. \begin{align*}x^2\end{align*}
7. \begin{align*}x^3\end{align*}
8. \begin{align*}\sqrt{x}\end{align*}
9. \begin{align*}y=\frac{2x+1}{x}\end{align*}
10. \begin{align*}y=\frac{x}{2x+1}\end{align*}
11. “The functions \begin{align*}f(x)\end{align*} and \begin{align*}g(x)\end{align*} are inverse functions.” Express that sentence in math, instead of in words (or using as few words as possible).
TAPPS Exercise: How Do I Solve That For y?
OK, so you’re looking for the inverse function of \begin{align*}y=\frac{x}{2x+1}\end{align*}. So you reverse the \begin{align*}x\end{align*} and the \begin{align*}y\end{align*} and you come up with \begin{align*}x=\frac{y}{2y+1}\end{align*}. Now you have to solve that for \begin{align*}y\end{align*}, and you’re stuck.
First of all, let’s review what that means! To “solve it for \begin{align*}y\end{align*}” means that we have to get it in the form \begin{align*}y=\end{align*} something, where the something has no \begin{align*}y\end{align*} in it anywhere. So \begin{align*}y=2x+4\end{align*} is solved for \begin{align*}y\end{align*}, but \begin{align*}y=2x+3y\end{align*} is not. Why? Because in the first case, if I give you \begin{align*}x\end{align*}, you can immediately find \begin{align*}y\end{align*}. But in the second case, you cannot.
“Solving it for \begin{align*}y\end{align*}” is also sometimes called “isolating \begin{align*}y\end{align*}” because you are getting \begin{align*}y\end{align*} all alone.
So that’s our goal. How do we accomplish it?
\begin{align*}x=\frac{y}{2y+1}\end{align*}
1. The biggest problem we have is the fraction. To get rid of it, we multiply both sides by \begin{align*}2y+1\end{align*}.
\begin{align*}x(2y+1)=y\end{align*}
2. Now, we distribute through. \begin{align*}2xy+x=y\end{align*}
3. Remember that our goal is to isolate \begin{align*}y\end{align*}. So now we get all the things with \begin{align*}y\end{align*} on one side, and all the things without \begin{align*}y\end{align*} on the other side.
\begin{align*}x=y-2xy\end{align*}
4. Now comes the key step: we factor out a \begin{align*}y\end{align*} from all the terms on the right side. This is the distributive property (like we did in step 2) done in reverse, and you should check it by distributing through.
\begin{align*}x=y(1-2x)\end{align*}
5. Finally, we divide both sides by what is left in the parentheses!
\begin{align*}\frac{x}{1-2x}=y\end{align*}
Ta-da! We’re done! \begin{align*}\frac{x}{1-2x}\end{align*} is the inverse function of \begin{align*}\frac{x}{2x+1}\end{align*}. Not convinced? Try two tests.
Test 1:
Test 2:
Now, you try it! Follow the above steps one at a time. You should switch roles at this point: the previous student should do the work, explaining each step to the previous teacher. Your job: find the inverse function of \begin{align*}y=\frac{x+1}{x-1}\end{align*}.
Name: __________________
Sample Test: Functions II
1. Joe and Lisa are baking cookies. Every cookie is a perfect circle. Lisa is experimenting with cookies of different radii (*the plural of “radius”). Unknown to Lisa, Joe is very competitive about his baking. He sneaks in to measure the radius of Lisa’s cookies, and then makes his own cookies have a \begin{align*}2\end{align*}" bigger radius.
Let \begin{align*}L\end{align*} be the radius of Lisa’s cookies. Let \begin{align*}J\end{align*} be the radius of Joe’s cookies. Let \begin{align*}a\end{align*} be the area of Joe’s cookies.
a. Write a function \begin{align*}J(L)\end{align*} that shows the radius of Joe’s cookies as a function of the radius of Lisa’s cookies.
b. Write a function \begin{align*}a(J)\end{align*} that shows the area of Joe’s cookies as a function of their radius. (If you don’t know the area of a circle, ask me—this information will cost you \begin{align*}1\end{align*} point.)
c. Now, put them together into the function \begin{align*}a(J(L))\end{align*} that gives the area of Joe’s cookies, as a direct function of the radius of Lisa’s.
d. Using that function, answer the question: if Lisa settles on a 3" radius, what will the area of Joe's cookies be? First, write the question in function notation—then solve it.
e. Using the same function, answer the question: if Joe’s cookies end up \begin{align*}49\pi\;\mathrm{square \ inches}\end{align*} in area, what was the radius of Lisa’s cookies? First, write the question in function notation—then solve it.
2. Make up a word problem involving composite functions, and having something to do with superheroes.
a. Describe the scenario. Remember that it must have something that depends on something else that depends on still another thing. If you have described the scenario carefully, I should be able to guess what your variables will be and all the functions that relate them.
b. Carefully name and describe all three variables.
c. Write two functions. One relates the first variable to the second, and the other relates the second variable to the third.
d. Put them together into a composite function that shows me how to get directly from the third variable to the first variable.
e. Using a sample number, write a (word problem!) question and use your composite function to find the answer.
3. Here is the algorithm for converting the temperature from Celsius to Fahrenheit. First, multiply the Celsius temperature by \begin{align*}\frac{9}{5}\end{align*}. Then, add \begin{align*}32\end{align*}.
a. Write this algorithm as a mathematical function: Celsius temperature \begin{align*}(C)\end{align*} goes in, Fahrenheit temperature \begin{align*}(F)\end{align*} comes out. \begin{align*}F(C)=\underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
b. Write the inverse of that function.
c. Write a real-world word problem that you can solve by using that inverse function. (This does not have to be elaborate, but it has to show that you know what the inverse function does.)
d. Use the inverse function that you found in part (b) to answer the question you asked in part (c).
4. \begin{align*}f(x)= \sqrt{x+1} \ g(x)=\frac{1}{x}\end{align*} . For a-e, I am not looking for answers like \begin{align*}[g(x)]^2\end{align*}. Your answers should not have a \begin{align*}g\end{align*} or an \begin{align*}f\end{align*} in them, just a bunch of \begin{align*}"x"s\end{align*}.
a. \begin{align*}f(g(x))=\end{align*}
b. \begin{align*}g(f(x))=\end{align*}
c. \begin{align*}f(f(x))=\end{align*}
d. \begin{align*}g(g(x))=\end{align*}
e. \begin{align*}g(f(g(x)))=\end{align*}
f. What is the domain of \begin{align*}f(x)?\end{align*}
g. What is the domain of \begin{align*}g(x)?\end{align*}
5. \begin{align*}f(x)=20-x\end{align*}
a. What is the domain?
b. What is the inverse function?
c. Test your inverse function. (No credit for just the words “it works”—I have to see your test.)
6. \begin{align*}f(x)=3+\frac{x}{7}\end{align*}
a. What is the domain?
b. What is the inverse function?
c. Test your inverse function. (Same note as above.)
7. \begin{align*}f(x)= \frac{2x}{3x-4}\end{align*}
a. What is the domain?
b. What is the inverse function?
c. Test your inverse function. (Same note.)
8. For each of the following diagrams, indicate roughly what the slope is.
a.
b.
c.
9. \begin{align*}6x+3y=10\end{align*}
\begin{align*}y=mx+b\end{align*} format: ___________
Slope: ___________
\begin{align*}y-\end{align*}intercept: ___________
Graph it!
Extra credit: Two numbers have the peculiar property that when you add them, and when you multiply them, you get the same answer.
a. If one of the numbers is \begin{align*}5\end{align*}, what is the other number?
b. If one of the numbers is \begin{align*}x\end{align*}, what is the other number? (Your answer will be a function of \begin{align*}x\end{align*}.)
c. What number could \begin{align*}x\end{align*} be that would not have any possible other number to go with it?
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