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 $y = 5 x - 7$ becomes $f (x) = 5 x - 7 .$ 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.

Evaluate the function in function notation

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Exercise

Given the function $f (x) = 3 x - 4,$ evaluate the following statements. $f (3) and f (x) = 23$

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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) = 3 x - 4

Substitute

x = 3

f (3) = 3 \cdot 3 - 4

MultiplyMultiply

f (3) = 9 - 4

SubTermSubtract term

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) = 3 x - 4

Substitute

f (x) = 23

23 = 3 x - 4

AddEqn

LHS + 4 = RHS + 4

27 = 3 x

DivEqn

LHS / 3 = RHS / 3

9 = x

RearrangeEqnRearrange equation

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

For which value of x is the function ’ s value 25 ?

Interpret the function notation

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Exercise

Julianne and Douglas drive from California to New Mexico. During the first four hours of the trip, the function $d (t) = 50 t$ 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. $d (3.5) and d (t) = 110$

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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.