Evaluating and Interpreting Function Notation

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Concept

Function Notation

Function notation is a special way to write functions that explicitly shows that yy is a function of x.x. In other words, that yy depends on x.x. Function notation is symbolically expressed as f(x)=y, f(x)=y, and read FF of xx equals y.y. Remember that xx represents the inputs of the function and yy represents the outputs. Written in function notation, the function y=5x7y=5x-7 becomes f(x)=5x7. f(x)=5x-7. Letters other than ff can be used to name a function. Additionally, function notation can be adjusted when the variable used to represent the input is not x.x. For example, a function describing how the value, V,V, of a car changes over time, t,t, can be expressed as

V(t). V(t).
Method

Evaluating Functions in Function Notation

Evaluating a function means finding the corresponding xx- or yy-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.
Exercise

Given the function f(x)=3x4,f(x)=3x-4, evaluate the following statements. f(3)andf(x)=23 f(3) \quad \text{and} \quad f(x)=23

Solution
Example

f(3)f(3)

To evaluate f(3)f(3) means to find the value of ff when x=3.x=3. To do this, we can substitute x=3x=3 into the rule and simplify.
f(x)=3x4f(x)=3x-4
f(3)=334f({\color{#0000FF}{3}})=3\cdot {\color{#0000FF}{3}}-4
f(3)=94f(3)=9-4
f(3)=5f(3)=5
The statement f(3)=5f(3)=5 means that when x=3,x=3, the function's value is 5.5.
Example

f(x)=23f(x)=23

When evaluating f(x)=23,f(x)=23, we want to find the value of xx for which ff equals 23.23. We begin by replacing f(x)f(x) with 23,23, then solve for xx using inverse operations.
f(x)=3x4f(x)=3x-4
23=3x4{\color{#0000FF}{23}}=3x-4
27=3x27=3x
9=x9=x
x=9x=9
Thus, f(9)=23.f(9)=23.
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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)=yf(x)=y mean. Suppose the following statement is given. f(5)=25 f(5)=25 The left-hand side, f(5),f(5), tells that the input of the function is x=5.x=5. The right-hand side, 25,25, means that for the given input value, the output of the function is 25.25. Additionally, the statement f(x)=25f(x)=25 asks

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

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

Solution
Example

d(3.5)d(3.5)

The function notation, d(t),d(t), gives d,d, the distance traveled in time t.t. When we write d(3.5), d(3.5), the time spent traveling is 3.53.5 hours. Thus, the statement as a whole gives the distance traveled at 3.53.5 hours.

Example

d(t)=110d(t)=110

Since d(t)d(t) gives the distance traveled in time, t,t, d(t)=110 d(t)=110 asks the number of hours it takes to travel a distance of 110110 miles.

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Exercises

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