Standards
MCC9-12.S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. ^{ ★ }
MCC9-12.S.ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. ^{ ★ }
MCC9-12.S.ID.6b Informally assess the fit of a function by plotting and analyzing residuals. ^{ ★ }
MCC9-12.S.ID.6c Fit a linear function for a scatter plot that suggests a linear association. ^{ ★ }
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 + 30,000\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? In this Concept, you'll learn how to convert equations such as this one to function notation and how to input a value into a function in order to get an output value.
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 \begin{align*}x\end{align*} there is one and only one value for \begin{align*}y\end{align*} .
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 \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.
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 \begin{align*}f(x)=-\frac{1}{2} x^2\end{align*} .
Evaluate \begin{align*}f(4)\end{align*} .
Solution: 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.
Example B
Rewrite \begin{align*}9x+3y=6\end{align*} using function notation.
Solution: 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*}
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, \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*} .
Example C
A function is defined as \begin{align*}f(x)=6x-36\end{align*} . Determine the following:
a) \begin{align*}f(2)\end{align*}
b) \begin{align*}f(p)\end{align*}
Solution:
a) 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*} .
b) Substitute \begin{align*}x = p\end{align*} into the function \begin{align*}f(x): \ f(p)=6p-36\end{align*} .
Guided Practice
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*} .
Solution:
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*}
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*}