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3.2 Domain and Range

Reading allowed inputs and outputs from an equation or graph, excluding division by zero and negative radicands, and graphing piecewise-defined rules.

Algebra and Trigonometry 2e, 3.2 Domain and Range

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35 exercises

3.2 Exercise 1

Find the domain of the following function: {(2, 10),(3, 10),(4, 20),(5, 30),(6, 40)}.

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3.2 Exercise 3

Find the domain of the function f(x)=x2-3f \left(x\right) = x^{2} - 3

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Finding the Domain of a Function as a Set of Ordered Pairs

Find the domain of the following function: { ( 2,10 ),( 3,10 ),( 4,20 ),( 5,30 ),( 6,40 ) } \left\{\left(2 , 10\right) , \left(3 , 10\right) , \left(4 , 20\right) , \left(5 , 30\right) , \left(6 , 40\right)\right\} .

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3.2 Exercise 13

Find the domain of the function:

{ (−5,4),(0,0),(5,−4),(10,−8),(15,−12) } \left\{\left(−5 , 4\right) , \left(0 , 0\right) , \left(5 , −4\right) , \left(10 , −8\right) , \left(15 , −12\right)\right\}

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Finding the Domain of a Function

Find the domain of the function f(x)= x 2 −1. f \left(x\right) = x^{2} - 1.

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3.2 Exercise 15

Find the domain of the function: f(x)=5−x+ x 3 . f \left(x\right) = 5 - x + x^{3} .

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3.2 Exercise 17

Find the domain of the function: f(x)= 1+4x 2x−1 . f \left(x\right) = \frac{1 + 4 x}{2 x - 1} .

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Finding the Domain of a Function with an Even Root

Find the domain of the function f(x)= 7−x . f \left(x\right) = \sqrt{7 - x} .

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3.2 Exercise 19

Find the domain of the function f(x)= 5+2x . f \left(x\right) = \sqrt{5 + 2 x} .

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Finding the Domain and Range Using Toolkit Functions

Find the domain and range of f(x)=2 x 3 −x. f \left(x\right) = 2 x^{3} - x .

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Finding the Domain and Range

Find the domain and range of f(x)= 2 x+1 . f \left(x\right) = \frac{2}{x + 1} .

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Finding the Domain and Range

Find the domain and range of f(x)=2 x+4 . f \left(x\right) = 2 \sqrt{x + 4} .

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3.2 Exercise 28

Find the domain and range of f( x )=− 2−x . f \left(x\right) = - \sqrt{2 - x} .

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Writing a Piecewise Function

A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, n, n , to the cost, C. C . Since one cannot have fractions of a person, this is really a discrete function. However, for this exercise we will treat it as a continuous function.

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Working with a Piecewise Function

A cell phone company uses the function below to determine the cost, C, C , in dollars for g g gigabytes of data transfer.

Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

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3.2 Exercise 33

Why does the domain differ for different functions?

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3.2 Exercise 35

Explain why the domain of f(x)= x 3 f \left(x\right) = \sqrt[3]{x} is different from the domain of f(x)= x . f \left(x\right) = \sqrt[]{x} .

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3.2 Exercise 37

How do you graph a piecewise function?

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3.2 Exercise 39

Find the domain of each function using interval notation.

f(x)=5−2 x 2 f \left(x\right) = 5 - 2 x^{2}

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3.2 Exercise 41

Find the domain of each function using interval notation.

f( x )=3− 6−2x f \left(x\right) = 3 - \sqrt{6 - 2 x}

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3.2 Exercise 43

Find the domain of each function using interval notation.

f(x)= x 2 +4 f \left(x\right) = \sqrt[]{x^{2} + 4}

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3.2 Exercise 45

Find the domain of each function using interval notation.

f(x)= x−1 3 f \left(x\right) = \sqrt[3]{x - 1}

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3.2 Exercise 47

Find the domain of each function using interval notation.

f( x )= 3x+1 4x+2 f \left(x\right) = \frac{3 x + 1}{4 x + 2}

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3.2 Exercise 49

Find the domain of each function using interval notation.

f(x)= x−3 x 2 +9x−22 f \left(x\right) = \frac{x - 3}{x^{2} + 9 x - 22}

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3.2 Exercise 51

Find the domain of each function using interval notation.

f(x)= 2 x 3 −250 x 2 −2x−15 f \left(x\right) = \frac{2 x^{3} - 250}{x^{2} - 2 x - 15}

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3.2 Exercise 53

Find the domain of each function using interval notation.

f(x)= 2x+1 5−x f \left(x\right) = \frac{2 x + 1}{\sqrt{5 - x}}

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3.2 Exercise 55

Find the domain of each function using interval notation.

f(x)= x−6 x−4 f \left(x\right) = \frac{\sqrt{x - 6}}{\sqrt{x - 4}}

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3.2 Exercise 57

Find the domain of each function using interval notation.

f(x)= x 2 −9x x 2 −81 f \left(x\right) = \frac{x^{2} - 9 x}{x^{2} - 81}

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3.2 Exercise 79

Given each function f,f , evaluate f(−3),f(−2),f(−1), f \left(−3\right) , f \left(−2\right) , f \left(−1\right) , and f(0). f \left(0\right) .

f(x)={ 1 if x≤−3 0 if x>−3 f \left(x\right) = \left\{\right. \begin{matrix}1 & \text{if}\textrm{ } x \leq - 3 \\ 0 & \text{if}\textrm{ } x > - 3\end{matrix}

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3.2 Exercise 81

Given each function f, f , evaluate f(−1),f(0),f(2), f \left(−1\right) , f \left(0\right) , f \left(2\right) , and f(4). f \left(4\right) .

f(x)={ 7x+3 if x<0 7x+6 if x≥0 f \left(x\right) = \left\{\right. \begin{matrix}7 x + 3 & \text{if} & x < 0 \\ 7 x + 6 & \text{if} & x \geq 0\end{matrix}

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3.2 Exercise 83

Given each function f, f , evaluate f(−1),f(0),f(2), f \left(−1\right) , f \left(0\right) , f \left(2\right) , and f(4). f \left(4\right) .

f( x )={ 5x if x<0 3 if 0≤x≤3 x 2 if x>3 f \left(x\right) = \left\{\right. \begin{matrix}5 x & \text{if} & x < 0 \\ 3 & \text{if} & 0 \leq x \leq 3 \\ x^{2} & \text{if} & x > 3\end{matrix}

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3.2 Exercise 85

Write the domain for the piecewise function in interval notation.

f(x)={ x 2 −2ifx<1 − x 2 +2ifx>1 f \left(x\right) = \left\{\right. \begin{matrix}x^{2} - 2 \text{if} x < 1 \\ - x^{2} + 2 \text{if} x > 1\end{matrix}

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3.2 Exercise 89

Suppose the range of a function f f is [−5,8]. \left[−5 , 8\right] . What is the range of |f(x)|? \left|f \left(x\right)\right| ?

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3.2 Exercise 91

Create a function in which the domain is x>2. x > 2.

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3.2 Exercise 93

The cost in dollars of making x x items is given by the function C(x)=10x+500. C \left(x\right) = 10 x + 500.

  1. ⓐThe fixed cost is determined when zero items are produced. Find the fixed cost for this item.
  2. ⓑWhat is the cost of making 25 items?
  3. ⓒSuppose the maximum cost allowed is $1500. What are the domain and range of the cost function, C(x)? C \left(x\right) ?
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Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.