# 4.13: Function Notation and Linear Functions

**Basic**Created by: CK-12

**Practice**Function Notation and Linear Functions

Suppose you just purchased a used car, and the number of miles on the odometer can be represented by the equation

### Guidance

So far, the term **function** has been used to describe many of the equations we have been graphing. The concept of a function is extremely important in mathematics. Not all equations are functions. To be a function, for each value of

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

Recall from a previous Concept that a function rule replaces the variable

If the function looks like this:

**Using Function Notation**

Function notation allows you to easily see the input value for the independent variable inside the parentheses.

#### Example A

Consider the function

Evaluate

**Solution:** The value inside the parentheses is the value of the variable

To use function notation, the equation must be written in terms of

#### Example B

Rewrite

**Solution:** The goal is to rearrange this equation so the equation looks like

**Functions as Machines**

You can think of a function as a machine. You start with an input (some value), the machine performs the operations (it does the work), and your output is the answer. For example, ** some number**,

When you use the function machine to evaluate

#### Example C

A function is defined as

a)

b)

**Solution:**

a) Substitute

b) Substitute

### Guided Practice

Rewrite the equation

**Solution:**

First we need to solve for

Adding

Now we just replace the

Now we can evaluate \begin{align*}f(x)=y=2x+5\end{align*} for \begin{align*}f(-1), f(2), f(0)\end{align*}, and \begin{align*}f(z)\end{align*}:

\begin{align*}f(-1)=2(-1)+5=-2+5=3\end{align*}

\begin{align*}f(2)=2(2)+5=4+5=9\end{align*}

\begin{align*}f(0)=2(0)+5=5\end{align*}

\begin{align*}f(z)=2z+5\end{align*}

### Practice

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: Linear Function Graphs (11:49)

(http://www.youtube.com/watch?v=EmTvdKkAUtE)

- How is \begin{align*}f(x)\end{align*} read?
- What does function notation allow you to do? Why is this helpful?
- Define
*function*. How can you tell if a graph is a function?

In 4 – 7, tell whether the graph is a function. Explain your reasoning.

Rewrite each equation using function notation.

- \begin{align*}y=7x-21\end{align*}
- \begin{align*}6x+8y=36\end{align*}
- \begin{align*}x=9y+3\end{align*}
- \begin{align*}y=6\end{align*}
- \begin{align*}d=65t+100\end{align*}
- \begin{align*}F=1.8C+32\end{align*}
- \begin{align*}s=0.10(m)+25,000\end{align*}

In 15 – 19, evaluate \begin{align*}f(-3), f(7), f(0)\end{align*}, and \begin{align*}f(z)\end{align*}.

- \begin{align*}f(x)=-2x+3\end{align*}
- \begin{align*}f(x)=0.7x+3.2\end{align*}
- \begin{align*}f(x)=\frac{5(2-x)}{11}\end{align*}
- \begin{align*}f(t)=\frac{1}{2} t^2+4\end{align*}
- \begin{align*}f(x)=3-\frac{1}{2} x\end{align*}

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### Image Attributions

Here you'll learn how to convert an equation to function notation and how to input a value into a function in order to get an output value.