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Mathematics LibreTexts

1.2E: Exercises

This page is a draft and is under active development. 

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1.2E: Exercises

Verbal

Exercise 1.2.1

Why does the domain differ for different functions?

Answer:
The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

Exercise 1.2.2

How do we determine the domain of a function defined by an equation?

Exercise 1.2.3

Explain why the domain of f(x)=3x is different from the domain of f(x)=x.

Answer:
There is no restriction on x for f(x)=3x because you can take the cube root of any real number. So the domain is all real numbers, (,). When dealing with the set of real numbers, you cannot take the square root of negative numbers. So x-values are restricted for f(x)=x to nonnegative numbers and the domain is [0,).

Exercise 1.2.4

When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

Exercise 1.2.5

How do you graph a piecewise function?

Answer:
Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the x-axis and y-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate or . Combine the graphs to find the graph of the piecewise function.

Algebraic

For the following exercises, find the domain of each function using interval notation.

Exercise 1.2.6

f(x)=2x(x1)(x2)

Exercise 1.2.7

f(x)=52x2

Answer:
(,)

Exercise 1.2.8

f(x)=3x2

Exercise 1.2.9

f(x)=362x

Answer:
(,3]

Exercise 1.2.10

f(x)=43x

Exercise 1.2.11

f(x)=x2+4

Answer:
(,)

Exercise 1.2.12

f(x)=312x

Exercise 1.2.13

f(x)=3x1

Answer:
(,)

Exercise 1.2.14

f(x)=9x6

Exercise 1.2.15

f(x)=3x+14x+2

Answer:
(,12)(12,)

Exercise 1.2.16

f(x)=x+4x4

Exercise 1.2.17

f(x)=x3x2+9x22

Answer:
(,11)(11,2)(2,)

Exercise 1.2.18

f(x)=1x2x6

Exercise 1.2.19

f(x)=2x3250x22x15

Answer:
(,3)(3,5)(5,)

Exercise 1.2.20

5x3

Exercise 1.2.21

2x+15x

Answer:
(,5)

Exercise 1.2.22

x4x6

Exercise 1.2.23

x6x4

Answer:
[6,)

Exercise 1.2.24

f(x)=xx

Exercise 1.2.25

f(x)=x29xx281

Answer:
(,9)(9,9)(9,)

Exercise 1.2.26

Find the domain of the function f(x)=2x350x by:

a. using algebra

b. graphing the function in the radicand and determining intervals on the x-axis for which the radicand is nonnegative..

Graphical

For the following exercises, write the domain and range of each function using interval notation.

Exercise 1.2.27

Graph of a function from \(\left(2, 8\right]\).

Answer:
domain: (2,8], range [6,8)

Exercise 1.2.28

Graph of a function from \(\left[4, 8\right)\).

Exercise 1.2.29

Graph of a function [-4,4]

Answer:
domain: [4,4], range: [0,2]

Exercise 1.2.30

Graph of a function [2,6]

Exercise 1.2.31

Graph of a function [-5,3)

Answer:
domain: [5,3), range: [0,2]

Exercise 1.2.32

Graph of a function from [-3, 2).

Exercise 1.2.33

Graph of a function from (-infinity, 2].

Answer:
domain: (,1], range: [0,)

Exercise 1.2.34

Graph of a function from [-4, infinity).

Exercise 1.2.35

Graph of a function from [-6, -1/6]U[1/6, 6]/.

Answer:
domain: [6,16][16,6]; range: [6,16][16,6]

Exercise 1.2.36

Graph of a function from (-2.5, infinity).

Exercise 1.2.37

Graph of a function from [-3, infinity).

Answer:
domain: [3,); range: [0,)

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

Exercise 1.2.38

f(x)={x+1if x<22x3if x2

Exercise 1.2.39

f(x)={2x1if x<11+xif x1

Answer:

domain: (,)

Graph of f(x).

Exercise 1.2.40

f(x)={x+1if x<0x1if x>0

Exercise 1.2.41

f(x)={3if x<0xif x0

Answer:

domain: (,)

Graph of f(x).

Exercise 1.2.42

f(x)={x2if x<01xif x>0

Exercise 1.2.43

f(x)={x2if x<0x+2if x0

Answer:

domain: (,)

Graph of f(x).

Exercise 1.2.44

f(x)={x+1if x<1x3if x1

Exercise 1.2.45

f(x)={|x|if x<21if x2

Answer:

domain: (,)

Graph of f(x).

Numeric

For the following exercises, given each function f, evaluate f(3), f(2), f(1), and f(0).

Exercise 1.2.46

f(x)={x+1if x<22x3if x2

Exercise 1.2.47

f(x)={1if x30if x>3

Answer:
f(3)=1; f(2)=0; f(1)=0; f(0)=0

Exercise 1.2.48

f(x)={2x2+3if x15x7if x>1

For the following exercises, given each function f, evaluate f(1), f(0), f(2), and f(4).

Exercise 1.2.49

f(x)={7x+3if x<07x+6if x0

Answer:
f(1)=4; f(0)=6; f(2)=20; f(4)=34

Exercise 1.2.50

f(x)={x22if x<24+|x5|if x2

Exercise 1.2.51

f(x)={5xif x<03if 0x2x2if x>3

Answer:
f(1)=5; f(0)=3; f(2)=3; f(4)=16

For the following exercises, write the domain for the piecewise function in interval notation.

Exercise 1.2.52

f(x)={x+1if x<22x3if x2

Exercise 1.2.53

f(x)={x22if x<1x2+2if x>1

Answer:
domain: (,1)(1,)

Exercise 1.2.54

f(x)={x23if x<03x2if x2

Technology

Exercise 1.2.55

Graph y=1x2 on the viewing window [0.5,0.1] and [0.1,0.5]. Determine the corresponding range for the viewing window. Show the graphs.

Answer:

Graph of the equation from [-0.5, -0.1].

window: [0.5,0.1]; range: [4,100]

Graph of the equation from [0.1, 0.5].

window: [0.1,0.5]; range: [4,100]

Exercise 1.2.56

Graph y=1x on the viewing window [0.5,0.1] and [0.1,0.5]. Determine the corresponding range for the viewing window. Show the graphs.

Extension

Exercise 1.2.57

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

Answer:
[0,8]

Exercise 1.2.58

Create a function in which the range is all nonnegative real numbers.

Exercise 1.2.59

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

Answer:
Many answers. One function is f(x)=1x2.

Real-World Applications

Exercise 1.2.60

The height h of a projectile is a function of the time t it is in the air. The height in feet for t seconds is given by the function h(t)=16t2+96t. What is the domain of the function? What does the domain mean in the context of the problem?

Answer:
The domain is [0,6]; it takes 6 seconds for the projectile to leave the ground and return to the ground.

Exercise 1.2.61

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

a. The fixed cost is determined when zero items are produced. Find the fixed cost for this item.

b. What is the cost of making 25 items?

c. Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function, C(x)?


1.2E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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1.2: Domain and Range
1.3: Rates of Change and Behavior of Graphs