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1.5: Functions as Rules and Tables

Difficulty Level: At Grade Created by: CK-12
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Instead of purchasing a one-day ticket to the theme park, Joseph decided to pay by ride. Each ride costs $2.00. To describe the amount of money Joseph will spend, several mathematical concepts can be used.

First, an expression can be written to describe the relationship between the cost per ride and the number of rides, \begin{align*}r\end{align*}. An equation can also be written if the total amount he wants to spend is known. An inequality can be used if Joseph wanted to spend less than a certain amount.

Example 1: Using Joseph’s situation, write the following:

a. An expression representing his total amount spent

b. An equation that shows Joseph wants to spend exactly $22.00 on rides

c. An inequality that describes the fact that Joseph will not spend more than $26.00 on rides

Solution: The variable in this situation is the number of rides Joseph will pay for. Call this \begin{align*}r\end{align*}.

a. \begin{align*}2(r)\end{align*}

b. \begin{align*}2(r) = 22\end{align*}

c. \begin{align*}2(r) \le 26\end{align*}

In addition to an expression, equation, or inequality, Joseph’s situation can be expressed in the form of a function or a table.

Definition: A function is a relationship between two variables such that the input value has ONLY one output value.

Writing Equations as Functions

A function is a set of ordered pairs in which the first coordinate, usually \begin{align*}x\end{align*}, matches with exactly one second coordinate, \begin{align*}y\end{align*}. Equations that follow this definition can be written in function notation. The \begin{align*}y\end{align*} coordinate represents the dependent variable, meaning the values of this variable depend upon what is substituted for the other variable.

Consider Joseph’s equation \begin{align*}m = 2r\end{align*}. Using function notation, the value of the equation (the money spent \begin{align*}m\end{align*}) is replaced with \begin{align*}f(r)\end{align*}. \begin{align*}f\end{align*} represents the function name and \begin{align*}(r)\end{align*} represents the variable. In this case the parentheses do not mean multiplication; they separate the function name from the independent variable.

\begin{align*}& \quad \ input\\ & \quad \ \ \ \downarrow\\ & \quad \underbrace{f(x)}= y \leftarrow output\\ & \ function\\ & \quad \ \ box\end{align*}

Example 2: Rewrite the following equations in function notation.

a. \begin{align*}y=7x-3\end{align*}

b. \begin{align*}d=65t\end{align*}

c. \begin{align*}F=1.8C+32\end{align*}


a. According to the definition of a function, \begin{align*}y=f(x)\end{align*}, so \begin{align*}f(x)=7x-3\end{align*}.

b. This time the dependent variable is \begin{align*}d\end{align*}. Function notation replaces the dependent variable, so \begin{align*}d = f(t) = 65t\end{align*}.

c. \begin{align*}F = f(C) = 1.8C + 32\end{align*}

Why Use Function Notation?

Why is it necessary to use function notation? The necessity stems from using multiple equations. Function notation allows one to easily decipher between the equations. Suppose Joseph, Lacy, Kevin, and Alfred all went to the theme park together and chose to pay $2.00 for each ride. Each person would have the same equation \begin{align*}m=2r\end{align*}. Without asking each friend, we could not tell which equation belonged to whom. By substituting function notation for the dependent variable, it is easy to tell which function belongs to whom. By using function notation, it will be much easier to graph multiple lines (Chapter 4).

Example 3: Write functions to represent the total each friend spent at the park.

Solution: \begin{align*}J(r)= 2r\end{align*} represents Joseph’s total, \begin{align*}L(r)= 2r\end{align*} represents Lacy's total, \begin{align*}K(r)= 2r\end{align*} represents Kevin's total, and \begin{align*}A(r)= 2r\end{align*} represents Alfred’s total.

Using a Function to Generate a Table

A function really is an equation. Therefore, a table of values can be created by choosing values to represent the independent variable. The answers to each substitution represent \begin{align*}f(x)\end{align*}.

Use Joseph’s function to generate a table of values. Because the variable represents the number of rides Joseph will pay for, negative values do not make sense and are not included in the value of the independent variable.

\begin{align*}R\end{align*} \begin{align*}J(r) = 2r\end{align*}
0 \begin{align*}2(0) = 0\end{align*}
1 \begin{align*}2(1) = 2\end{align*}
2 \begin{align*}2(2) = 4\end{align*}
3 \begin{align*}2(3) = 6\end{align*}
4 \begin{align*}2(4) = 8\end{align*}
5 \begin{align*}2(5) = 10\end{align*}
6 \begin{align*}2(6) = 12\end{align*}

As you can see, the list cannot include every possibility. A table allows for precise organization of data. It also provides an easy reference for looking up data and offers a set of coordinate points that can be plotted to create a graphical representation of the function. A table does have limitations; namely it cannot represent infinite amounts of data and it does not always show the possibility of fractional values for the independent variable.

Domain and Range of a Function

The set of all possible input values for the independent variable is called the domain. The domain can be expressed in words, as a set, or as an inequality. The values resulting from the substitution of the domain represent the range of a function.

The domain of Joseph’s situation will not include negative numbers because it does not make sense to ride negative rides. He also cannot ride a fraction of a ride, so decimals and fractional values do not make sense as input values. Therefore, the values of the independent variable r will be whole numbers beginning at zero.

Domain: All whole numbers

The values resulting from the substitution of whole numbers are whole numbers times two. Therefore, the range of Joseph’s situation is still whole numbers just twice as large.

Range: All even whole numbers

Example 4: A tennis ball is bounced from a height and bounces back to 75% of its previous height. Write its function and determine its domain and range.

Solution: The function of this situation is \begin{align*}h(b)= 0.75b\end{align*}, where \begin{align*}b\end{align*} represents the previous bounce height.

Domain: The previous bounce height can be any positive number, so \begin{align*}b \ge 0\end{align*}.

Range: The new height is 75% of the previous height, and therefore will also be any positive number (decimal or whole number), so the range is all positive real numbers.

Multimedia Link For another look at the domain of a function, see the following video where the narrator solves a sample problem from the California Standards Test about finding the domain of an unusual function. Khan Academy CA Algebra I Functions (6:34)

Write a Function Rule

In many situations, data is collected by conducting a survey or an experiment. To visualize the data, it is arranged into a table. Most often, a function rule is needed to predict additional values of the independent variable.

Example 5: Write a function rule for the table.

\begin{align*}& \text{Number of CDs} && 2 && 4 && 6 && 8 && 10\\ & \text{Cost}\ (\$) && 24 && 48 && 72 && 96 && 120\end{align*}

Solution: You pay $24 for 2 CDs, $48 for 4 CDs, and $120 for 10 CDs. That means that each CD costs $12.

We can write the function rule.

\begin{align*}\text{Cost} = \$12 \times \text{number of CDs}\end{align*} or \begin{align*}f(x) = 12x\end{align*}

Example 6: Write a function rule for the table.

\begin{align*}&&x && -3 && -2 && -1 && 0 && 1 && 2 && 3\\ &&y && \quad 3 && \quad 2 && \quad 1 && 0 && 1 && 2 && 3\end{align*}

Solution: The values of the dependent variable are always the positive outcomes of the input values. This relationship has a special name, the absolute value. The function rule looks like this: \begin{align*}f(x) = |x|\end{align*}.

Represent a Real-World Situation with a Function

Let’s look at a real-world situation that can be represented by a function.

Example 7: Maya has an internet service that currently has a monthly access fee of $11.95 and a connection fee of $0.50 per hour. Represent her monthly cost as a function of connection time.

Solution: Let \begin{align*}x=\end{align*} the number of hours Maya spends on the internet in one month and let \begin{align*}y=\end{align*} Maya’s monthly cost. The monthly fee is $11.95 with an hourly charge of $0.50.

The total cost \begin{align*}=\end{align*} flat fee \begin{align*}+\end{align*} hourly fee \begin{align*}\times\end{align*} number of hours. The function is \begin{align*}y = f(x) = 11.95 + 0.50x\end{align*}

Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both. CK-12 Basic Algebra: Domain and Range of a Function (12:52)

  1. Rewrite using function notation: \begin{align*}y= \frac{5}{6} x-2\end{align*}.
  2. What is one benefit of using function notation?
  3. Define domain.
  4. True or false? Range is the set of all possible inputs for the independent variable.
  5. Generate a table from \begin{align*}-5 \le x \le 5\end{align*} for \begin{align*}f(x)= -(x)^2- 2\end{align*}
  6. Use the following situation for question 6: Sheri is saving for her first car. She currently has $515.85 and is savings $62 each week.
    1. Write a function rule for the situation.
    2. Can the domain be “all real numbers"? Explain your thinking.
    3. How many weeks would it take Sheri to save $1,795.00?

In 7 - 11, identify the domain and range of the function.

  1. Dustin charges $10 per hour for mowing lawns.
  2. Maria charges $25 per hour for math tutoring, with a minimum charge of $15.
  3. \begin{align*}f(x) = 15x - 12\end{align*}
  4. \begin{align*}f(x) = 2x^2 + 5\end{align*}
  5. \begin{align*}f(x)=\frac{1}{x}\end{align*}
  6. What is the range of the function \begin{align*}y = x^2 - 5\end{align*} when the domain is \begin{align*}-2\end{align*}, \begin{align*}-1\end{align*}, 0, 1, 2?
  7. What is the range of the function \begin{align*}y = 2x - \frac{3}{4}\end{align*} when the domain is \begin{align*}-2.5\end{align*}, 1.5, 5?
  8. Angie makes $6.50 per hour working as a cashier at the grocery store. Make a table of values that shows her earning for the input values 5, 10, 15, 20, 25, 30.
  9. The area of a triangle is given by: \begin{align*}A = \frac{1}{2}bh\end{align*}. If the base of the triangle is 8 centimeters, make a table of values that shows the area of the triangle for heights 1, 2, 3, 4, 5, and 6 centimeters.
  10. Make a table of values for the function \begin{align*}f(x) = \sqrt{2x + 3}\end{align*} for the input values \begin{align*}-1\end{align*}, 0, 1, 2, 3, 4, 5.
  11. Write a function rule for the table.

\begin{align*}&&x && 3 && 4 && 5 && 6\\ &&y && 9 && 16 && 25 && 36\end{align*}

  1. Write a function rule for the table.

\begin{align*}&\text{hours} && 0 && 1 && 2 && 3\\ &\text{cost} && 15 && 20 && 25 && 30\end{align*}

  1. Write a function rule for the table.

\begin{align*}&&x && 0 && 1 && 2 && 3\\ &&y && 24 && 12 && 6 && 3\end{align*}

  1. Write a function that represents the number of cuts you need to cut a ribbon in \begin{align*}x\end{align*} number of pieces.
  2. Solomon charges a $40 flat rate and $25 per hour to repair a leaky pipe. Write a function that represents the total fee charged as a function of hours worked. How much does Solomon earn for a three-hour job?
  3. Rochelle has invested $2500 in a jewelry making kit. She makes bracelets that she can sell for $12.50 each. How many bracelets does Rochelle need to make before she breaks even?
  4. Make up a situation in which the domain is all real numbers but the range is all whole numbers.

Mixed Review

  1. Compare the following numbers \begin{align*}23 \ \underline{\;\;\;\;\;} \ 21.999\end{align*}.
  2. Write an equation to represent the following: the quotient of 96 and 4 is \begin{align*}g\end{align*}.
  3. Write an inequality to represent the following: 11 minus \begin{align*}b\end{align*} is at least 77.
  4. Find the value of the variable \begin{align*}k:13(k)=169\end{align*}.

Quick Quiz

1. Write a function rule to describe the following table:

\begin{align*}&\# \ \text{of Books} && 1 && 2 && 3 && 4 && 5 && 6\\ &\text{Cost} && 4.75 && 5.25 && 5.75 && 6.25 && 6.75 && 7.25\end{align*}

2. Simplify: \begin{align*}84 \div [(18-16) \times 3]\end{align*}.

3. Evaluate the expression \begin{align*}\frac{2}{3} (y+6)\end{align*} when \begin{align*}y=3\end{align*}.

4. Rewrite using function notation: \begin{align*}y = \frac{1}{4} x^2\end{align*}.

5. You purchased six video games for $29.99 each and three DVD movies for $22.99. What is the total amount of money you spent?

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