At the amusement park, Taylor noticed that there seemed to be a pattern for people who won the dart throwing game. She was so curious that she watched people play the game for a few hours. When 12 people played, there were only 6 winners. When ten people played, there were five winners.
This is a table to represent the data that Taylor collected.
Input | Output |
---|---|
12 | 6 |
10 | 5 |
8 | 4 |
6 | 3 |
4 | 2 |
Do you see a pattern?
What rule could we write to represent what happened to the input to equal the output?
This Concept is about writing and evaluating function rules. By the end of it, you will be able to help Taylor with this dilemma.
Guidance
In the last Concept, you were introduced to rules and input - output tables. You also heard the word function for the first time. Let's look at what a function is and then how we can evaluate a given function rule.
A function is when one variable or terms depends on another according to a rule. There is a special relationship between the two variables of the function where each value in the input applies to only one value in the output.
These rules that we have been writing we can call function rules, because they explain how the function operates. Here are some hints for writing function rules.
Hints for Writing Function Rules
- Decipher the pattern of the function. What happened to the input to get the output?
- Write the rule as an expression.
Think of the input as a variable.
Then write the operations used with this variable.
This will explain the function rule. In other words, the function rule is the same thing as the expression.
Now you will be given function rules and you must work to determine whether or not the rule is a rule for the table.
Is \begin{align*}x + 4\end{align*} a rule for this function?
Input | Output |
---|---|
2 | 5 |
3 | 6 |
4 | 7 |
5 | 8 |
No. It is not. Look at the input. Each term in the input became the term in the output when 3 was added to it.
Our rule states that four was added. Therefore, this is not a viable rule.
Is \begin{align*}5x\end{align*} a rule for this function?
Input | Output |
---|---|
20 | 100 |
10 | 50 |
5 | 25 |
1 | 5 |
Yes it is. In this case, each term in the input was multiplied by five to get the term in the output. Therefore this rule does work for this table.
Practice a few of these on your own. Figure out if each rule makes sense for the input-output table.
Example A
\begin{align*}4x\end{align*}
Input | Output |
---|---|
2 | 10 |
3 | 15 |
5 | 25 |
6 | 30 |
Solution: No, this rule does not work for this table.
Example B
\begin{align*}2x-1\end{align*}
Input | Output |
---|---|
2 | 3 |
3 | 5 |
4 | 7 |
6 | 11 |
Solution: Yes, this rule works for this table.
Example C
\begin{align*}3x\end{align*}
Input | Output |
---|---|
2 | 6 |
3 | 9 |
4 | 12 |
6 | 18 |
Solution: Yes, this rule works for this table.
Now back to Taylor and the amusement park. Here is the original problem once again.
At the amusement park, Taylor noticed that there seemed to be a pattern for people who won the dart throwing game. She was so curious that she watched people play the game for a few hours. When 12 people played, there were only 6 winners. When ten people played, there were five winners.
This is a table to represent the data that Taylor collected.
Input | Output |
---|---|
12 | 6 |
10 | 5 |
8 | 4 |
6 | 3 |
4 | 2 |
Do you see a pattern?
What rule could we write to represent what happened to the input to equal the output?
If you look, you will see that each term of the input was divided by two to get the output. We can use a variable for the input.
\begin{align*}\frac{a}{2}\end{align*}
This is our answer.
Vocabulary
Here are the vocabulary words in this Concept.
- Pattern
- a series of pictures, numbers or other symbols that repeats in some way according to rule.
- Function
- one variable depends on the other and there is only one output for each input in a function.
- Input-Output Table
- A table that shows how a value changes according to a rule.
Guided Practice
Here is one for you to try on your own.
Input | Output |
---|---|
3 | 5 |
5 | 9 |
7 | 13 |
8 | 15 |
10 | 19 |
What rule could we write to represent this function?
Answer
Here two operations were performed. The input value was multiplied by two and then one was subtracted. We can use a variable for the input and write the rule.
Rule: \begin{align*}2x-1\end{align*}
Video Review
Here are videos for review.
James Sousa, Introduction to Functions, Part 1
James Sousa, Introduction to Functions, Part 2
Practice
Directions: Evaluate each given function rule. Write yes if the rule works and no if the rule does not work.
1. \begin{align*}2x+2\end{align*}
Input | Output |
---|---|
2 | 6 |
3 | 8 |
4 | 10 |
5 | 12 |
2. \begin{align*}3x\end{align*}
Input | Output |
---|---|
1 | 4 |
2 | 6 |
3 | 10 |
3. \begin{align*}5x+1\end{align*}
Input | Output |
---|---|
1 | 6 |
2 | 11 |
3 | 16 |
4 | 21 |
4. \begin{align*}2x\end{align*}
Input | Output |
---|---|
1 | 3 |
2 | 5 |
3 | 7 |
5. \begin{align*}3x-1\end{align*}
Input | Output |
---|---|
1 | 2 |
2 | 5 |
3 | 8 |
4 | 11 |
6. \begin{align*}2x+1\end{align*}
Input | Output |
---|---|
1 | 3 |
2 | 4 |
3 | 6 |
5 | 10 |
7. \begin{align*}4x\end{align*}
Input | Output |
---|---|
0 | 0 |
1 | 4 |
2 | 8 |
3 | 12 |
8. \begin{align*}6x-3\end{align*}
Input | Output |
---|---|
1 | 3 |
2 | 9 |
3 | 15 |
9. \begin{align*}2x\end{align*}
Input | Output |
---|---|
0 | 0 |
1 | 2 |
2 | 4 |
3 | 6 |
10. \begin{align*}3x-3\end{align*}
Input | Output |
---|---|
1 | 0 |
2 | 3 |
4 | 9 |
5 | 12 |
Directions: Create a table for each rule.
11. \begin{align*}5x\end{align*}
12. \begin{align*}6x+1\end{align*}
13. \begin{align*}2x - 3\end{align*}
14. \begin{align*}3x+3\end{align*}
15. \begin{align*}4x+1\end{align*}