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# Function Notation and Linear Functions

## If y=mx+b and y=f(x) then f(x)=mx+b

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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 \begin{align*}y = x + 30000\end{align*}, where \begin{align*}y\end{align*} is the number of miles on the odometer, and \begin{align*}x\end{align*} is the number of miles you have driven it. Could you convert this equation to function notation? How many miles will be on the odometer if you drive the car 700 miles?

### Function Notation

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 \begin{align*}x\end{align*} there is one and only one value for \begin{align*}y\end{align*}.

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

Recall that a function rule replaces the variable \begin{align*}y\end{align*} with its function name, usually \begin{align*}f(x)\end{align*}. Remember that these parentheses do not mean multiplication. They separate the function name from the independent variable, \begin{align*}x\end{align*}.

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

\begin{align*}f(x)\end{align*} is read “the function \begin{align*}f\end{align*} of \begin{align*}x\end{align*}” or simply “\begin{align*}f\end{align*} of \begin{align*}x\end{align*}.”

If the function looks like this: \begin{align*}h(x)=3x-1\end{align*}, it would be read \begin{align*}h\end{align*} of \begin{align*}x\end{align*} equals 3 times \begin{align*}x\end{align*} minus 1.

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

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, \begin{align*}f(x)=3x+2\end{align*} takes some number, \begin{align*}x\end{align*}, multiplies it by 3 and adds 2. As a machine, it would look like this:

When you use the function machine to evaluate \begin{align*}f(2)\end{align*}, the solution is \begin{align*}f(2)=8\end{align*}.

#### Let's use function notation to complete the following problems:

1. Consider the function \begin{align*}f(x)=-\frac{1}{2} x^2\end{align*}. Evaluate \begin{align*}f(4)\end{align*}.

The value inside the parentheses is the value of the variable \begin{align*}x\end{align*}. Use the Substitution Property to evaluate the function for \begin{align*}x=4\end{align*}.

\begin{align*}f(4)& =-\frac{1}{2}(4^2)\\ f(4)& = -\frac{1}{2} \cdot 16\\ f(4)& =-8\end{align*}

To use function notation, the equation must be written in terms of \begin{align*}x\end{align*}. This means that the \begin{align*}y-\end{align*}variable must be isolated on one side of the equal sign.

1. Rewrite \begin{align*}9x+3y=6\end{align*} using function notation.

The goal is to rearrange this equation so the equation looks like \begin{align*}y=\end{align*}. Then replace \begin{align*}y=\end{align*} with \begin{align*}f(x)=\end{align*}.

\begin{align*}9x+3y& =6 && \text{Subtract} \ 9x \ \text{from both sides}.\\ 3y& =6-9x && \text{Divide by} \ 3.\\ y& =\frac{6-9x}{3}=2-3x\\ f(x)& =2-3x\end{align*}

1. A function is defined as \begin{align*}f(x)=6x-36\end{align*}. Determine \begin{align*}f(2)\end{align*} and \begin{align*}f(p)\end{align*}

Substitute \begin{align*}x = 2\end{align*} into the function \begin{align*}f(x): \ f(2)=6 \cdot 2 - 36 = 12-36=-24\end{align*}.

Substitute \begin{align*}x = p\end{align*} into the function \begin{align*}f(x): \ f(p)=6p-36\end{align*}.

### Examples

#### Example 1

Earlier, you were told that the number of miles on your used car can be represented by the equation \begin{align*}y = x + 30000\end{align*} where \begin{align*}y\end{align*} is the number of miles on the odometer and \begin{align*}x\end{align*} is the number of miles you have driven it. What is this equation converted into function notation? How many miles will be on the odometer if you drive the car 700 miles?

To write an equation in function notation, replace the \begin{align*}y\end{align*} variable with \begin{align*}f(x)\end{align*}. The equation written in function notation would be: \begin{align*}f(x)= x + 30000\end{align*} where \begin{align*}f(x)\end{align*} is the number of miles on the odometer.

To figure out how many miles will be on the odometer if you drive the car 700 miles, we want to find \begin{align*}f(700)\end{align*}.

\begin{align*}f(700) = 700 + 30000\\ f(700)=30700\end{align*}

There will be 30,700 miles on the odometer if you drive the car 700 miles.

#### Example 2

Rewrite the equation \begin{align*}2y-4x=10\end{align*} in function notation where \begin{align*}f(x)=y\end{align*}, and then evaluate \begin{align*}f(-1), f(2), f(0)\end{align*}, and \begin{align*}f(z)\end{align*}.

First we need to solve for \begin{align*}y\end{align*}.

Adding \begin{align*}4x\end{align*} to both sides gives \begin{align*}2y=4x+10\end{align*}, and dividing by 2 gives \begin{align*}y=2x+5.\end{align*}

Now we just replace the \begin{align*}y\end{align*} with \begin{align*}f(x)\end{align*} to get \begin{align*}f(x)=2x+5.\end{align*}.

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*}

### Review

1. How is \begin{align*}f(x)\end{align*} read?
2. What does function notation allow you to do? Why is this helpful?
3. 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.

In 8-14, rewrite each equation using function notation.

1. \begin{align*}y=7x-21\end{align*}
2. \begin{align*}6x+8y=36\end{align*}
3. \begin{align*}x=9y+3\end{align*}
4. \begin{align*}y=6\end{align*}
5. \begin{align*}d=65t+100\end{align*}
6. \begin{align*}F=1.8C+32\end{align*}
7. \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*}.

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

To see the Review answers, open this PDF file and look for section 4.13.

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