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Evaluating and Interpreting Function Notation

Concept

Function Notation

Function notation is a special way to write functions that explicitly shows that $y$ is a function of $x.$ In other words, that $y$ depends on $x.$ Function notation is symbolically expressed as $f(x)=y,$ and read $f$ of $x$ equals $y.$ Remember that $x$ represents the inputs of the function and $y$ represents the outputs. Written in function notation, the function $y=5x-7$ becomes $f(x)=5x-7.$ Letters other than $f$ can be used to name a function. Additionally, function notation can be adjusted when the variable used to represent the input is not $x.$ For example, a function describing how the value, $V,$ of a car changes over time, $t,$ can be expressed as

$V(t).$
Method

Evaluating Functions in Function Notation

Evaluating a function means finding the corresponding $x$- or $y$-value, given the other. When the function is given as a function rule, this is done by substituting the given value and solving for the remaining variable.
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Exercise

Given the function $f(x)=3x-4,$ evaluate the following statements. $f(3) \quad \text{and} \quad f(x)=23$

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Solution
Example

$f(3)$

To evaluate $f(3)$ means to find the value of $f$ when $x=3.$ To do this, we can substitute $x=3$ into the rule and simplify.
$f(x)=3x-4$
$f({\color{#0000FF}{3}})=3\cdot {\color{#0000FF}{3}}-4$
$f(3)=9-4$
$f(3)=5$
The statement $f(3)=5$ means that when $x=3,$ the function's value is $5.$
Example

$f(x)=23$

When evaluating $f(x)=23,$ we want to find the value of $x$ for which $f$ equals $23.$ We begin by replacing $f(x)$ with $23,$ then solve for $x$ using inverse operations.
$f(x)=3x-4$
${\color{#0000FF}{23}}=3x-4$
$27=3x$
$9=x$
$x=9$
Thus, $f(9)=23.$
Explanation

What does function notation mean?

Interpreting statements in function notation is sometimes necessary. To accomplish this, it's important to understand what the left- and right-hand sides of $f(x)=y$ mean. Suppose the following statement is given. $f(5)=25$ The left-hand side, $f(5),$ tells that the input of the function is $x=5.$ The right-hand side, $25,$ means that for the given input value, the output of the function is $25.$ Additionally, the statement $f(x)=25$ asks

\begin{aligned} \text{For which value of} \ x \ \text{is the function's value} \ 25 \text{?} \end{aligned}
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Exercise

Julianne and Douglas drive from California to New Mexico. During the first four hours of the trip, the function $d(t)=50t$ describes $d,$ the distance in miles they've driven in $t,$ the number of hours they've been traveling. Interpret the meaning of the following statements. $d(3.5) \quad \text{and} \quad d(t)=110$

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Solution
Example

$d(3.5)$

The function notation, $d(t),$ gives $d,$ the distance traveled in time $t.$ When we write $d(3.5),$ the time spent traveling is $3.5$ hours. Thus, the statement as a whole gives the distance traveled at $3.5$ hours.

Example

$d(t)=110$

Since $d(t)$ gives the distance traveled in time, $t,$ $d(t)=110$ asks the number of hours it takes to travel a distance of $110$ miles.