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# Function Notation

## Explore f(x) notation for functions

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Function Notation

Suppose you worked at an animal rescue caring for the dogs, and you wanted to determine the age of each of the dogs in 'dog years', based on the age in human years. Since you would be performing the same calculation over and over, your friend suggests that you write a function. How would you go about setting up such a function? Is a function really different from a two variable, x = y form of an equation? In this Concept, you'll learn how to write functions.

### Function Notation

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 could be written to describe the relationship between the cost per ride and the number of rides, r\begin{align*}r\end{align*}. An equation could be written, if the total amount he wants to spend is known. An inequality could be used if Joseph wanted to spend less than a certain amount. #### Using Joseph’s situation, write the following: The variable in this situation is the number of rides Joseph will pay for. Call this r\begin{align*}r\end{align*}. a. An expression representing his total amount spent 2(r)\begin{align*}2(r)\end{align*} b. An equation representing his total amount spent 2(r)=m\begin{align*}2(r) = m\end{align*} c. An equation that shows Joseph wants to spend exactly22.00 on rides

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

#### Write functions to represent the total each friend spent at the park.

\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.

### Examples

Recall the example from a previous Concept where a student organization sells shirts to raise money. The cost of printing the shirts was expressed as \begin{align*}100+7x\end{align*} and for the revenue, we had the expression \begin{align*} 15x\end{align*}, where \begin{align*}x\end{align*} is the number of shirts.

#### Example 1

Write two functions, one for the cost and one for revenue.

The cost function we will write as \begin{align*}C(x)=100+7x\end{align*} and the revenue function we will write as \begin{align*}R(x)=15x\end{align*}.

#### Example 2

Express that the cost must be less than or equal to 800. Since \begin{align*}C(x)\end{align*} represents the costs, we substitute in800 for \begin{align*}C(x)\end{align*} and replace the equation with the appropriate inequality symbol

\begin{align*}100+7x \le 800\end{align*}

We want to find the value of \begin{align*}x\end{align*} that will make this equation true. It looks like 100 is the answer. Checking this (see below) it is clear that 100 does satisfy the equation. The students must sell 100 shirts in order to have a revenue of 1500. \begin{align*}1500 & =15(100)\\ 1500 & =1500\end{align*} Since 1500 = 1500 is a true statement, 100 shirts is the correct number. ### Review 1. Rewrite using function notation: \begin{align*}y= \frac{5}{6} x-2\end{align*}. 2. Rewrite using function notation: \begin{align*}m=n^2+2n-3\end{align*}. 3. What is one benefit of using function notation? 4. Write a function that expresses the money earned after working some number of hours for10 an hour.
5. Write a function that represents the number of cuts you need to cut a ribbon in \begin{align*}x\end{align*} number of pieces.
6. Jackie and Mayra each will collect a \$2 pledge for every basket they make during a game. Write two functions, one for each girl, expressing how much money she will collect.

To see the Review answers, open this PDF file and look for section 1.10.

### Vocabulary Language: English Spanish

dependent variable

A dependent variable is one whose values depend upon what is substituted for the other variable.

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.

independent variable

The independent variable is the variable which is not dependent on another variable. The dependent variable is dependent on the independent variable.