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# 1.2E: Exercises

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## 1.2E: Exercises

Verbal

Exercise 1.2.1

Why does the domain differ for different functions?

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 $$f(x)=\sqrt{x}$$ is different from the domain of $$f(x)=\sqrt{x}$$.

There is no restriction on x for $$f(x)=\sqrt{x}$$ 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 $$f(x)=\sqrt{x}$$ 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?

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

$$f(x)=−2x(x−1)(x−2)$$

Exercise 1.2.7

$$f(x)=5−2x^2$$

$$(-\infty,\infty)$$

Exercise 1.2.8

$$f(x)=3\sqrt{x-2}$$

Exercise 1.2.9

$$f(x)=3−\sqrt{6−2x}$$

$$\left(-\infty,3\right]$$

Exercise 1.2.10

$$f(x)=\sqrt{4-3x}$$

Exercise 1.2.11

$$f(x)=\sqrt{x^2+4}$$

$$(-\infty,\infty)$$

Exercise 1.2.12

$$f(x)=\sqrt{1-2x}$$

Exercise 1.2.13

$$f(x)=\sqrt{x-1}$$

$$(-\infty,\infty)$$

Exercise 1.2.14

$$f(x)=\dfrac{9}{x-6}$$

Exercise 1.2.15

$$f(x)=\dfrac{3x+1}{4x+2}$$

$$(-\infty,-\dfrac{1}{2})\cup(-\dfrac{1}{2},\infty)$$

Exercise 1.2.16

$$f(x)=\dfrac{\sqrt{x+4}}{x-4}$$

Exercise 1.2.17

$$f(x)=\dfrac{x-3}{x^2+9x-22}$$

$$(-\infty,-11)\cup(-11,2)\cup(2,\infty)$$

Exercise 1.2.18

$$f(x)=\dfrac{1}{x^2-x-6}$$

Exercise 1.2.19

$$f(x)=\dfrac{2x^3−250}{x^2−2x−15}$$

$$(-\infty,-3)\cup(-3,5)\cup(5,\infty)$$

Exercise 1.2.20

$$\dfrac{5}{\sqrt{x-3}}$$

Exercise 1.2.21

$$\dfrac{2x+1}{\sqrt{5-x}}$$

$$(-\infty,5)$$

Exercise 1.2.22

$$\dfrac{\sqrt{x-4}}{\sqrt{x-6}}$$

Exercise 1.2.23

$$\dfrac{\sqrt{x-6}}{\sqrt{x-4}}$$

$$\left[6,\infty\right)$$

Exercise 1.2.24

$$f(x)=\dfrac{x}{x}$$

Exercise 1.2.25

$$f(x)=\dfrac{x^2-9x}{x^2-81}$$

$$(-\infty,-9)\cup(-9,9)\cup(9,\infty)$$

Exercise 1.2.26

Find the domain of the function $$f(x)=\sqrt{2x^3-50x}$$ 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 domain: $$\left(2,8\right]$$, range $$\left[6,8\right)$$

Exercise 1.2.28 Exercise 1.2.29 domain: $$[−4, 4]$$, range: $$[0, 2]$$

Exercise 1.2.30 Exercise 1.2.31 domain: $$\left[−5, 3\right)$$, range: $$[0,2]$$

Exercise 1.2.32 Exercise 1.2.33 domain: $$\left(−\infty,1\right]$$, range: $$\left[0,\infty\right)$$

Exercise 1.2.34 Exercise 1.2.35 domain: $$\left[−6,−\frac{1}{6}\right]\cup\left[\frac{1}{6},6\right]$$; range: $$\left[−6,−\frac{1}{6}\right]\cup\left[\frac{1}{6},6\right]$$

Exercise 1.2.36 Exercise 1.2.37 domain: $$\left[−3, \infty\right)$$; range: $$\left[0,\infty\right)$$

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

Exercise 1.2.38

$$f(x)= \begin{cases} x+1 & \text{if x < -2} \\ -2x-3 & \text{if x {\geq} -2} \end{cases}$$

Exercise 1.2.39

$$f(x)= \begin{cases} 2x-1 & \text{if x < 1} \\ 1+x & \text{if x {\geq} 1} \end{cases}$$

domain: $$(−\infty,\infty)$$ Exercise 1.2.40

$$f(x)= \begin{cases} x+1 & \text{if x < 0} \\ x-1 & \text{if x > 0} \end{cases}$$

Exercise 1.2.41

$$f(x)= \begin{cases} 3 & \text{if x < 0} \\ \sqrt{x} & \text{if x {\geq} 0} \end{cases}$$

domain: $$(−\infty,\infty)$$ Exercise 1.2.42

$$f(x)= \begin{cases} x^2 & \text{if x < 0} \\ 1-x & \text{if x > 0} \end{cases}$$

Exercise 1.2.43

$$f(x)= \begin{cases} x^2 & \text{if x < 0} \\ x+2 & \text{if x {\geq} 0} \end{cases}$$

domain: $$(−\infty,\infty)$$ Exercise 1.2.44

$$f(x)= \begin{cases} x+1 & \text{if x < 1} \\ x^3 & \text{if x {\geq} 1} \end{cases}$$

Exercise 1.2.45

$$f(x)= \begin{cases} |x| & \text{if x < 2} \\ 1 & \text{if x {\geq} 2} \end{cases}$$

domain: $$(−\infty,\infty)$$ Numeric

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

Exercise 1.2.46

$$f(x)= \begin{cases} x+1 & \text{if x < -2} \\ -2x-3 & \text{if x {\geq} -2} \end{cases}$$

Exercise 1.2.47

$$f(x)= \begin{cases} 1 & \text{if x \leq -3} \\ 0 & \text{if x > -3} \end{cases}$$

$$f(−3)=1$$; $$f(−2)=0$$; $$f(−1)=0$$; $$f(0)=0$$

Exercise 1.2.48

$$f(x)= \begin{cases} -2x^2+3 & \text{if x \leq -1} \\ 5x-7 & \text{if x > -1} \end{cases}$$

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

Exercise 1.2.49

$$f(x)= \begin{cases} 7x+3 & \text{if x < 0} \\ 7x+6 & \text{if x {\geq} 0} \end{cases}$$

$$f(−1)=−4$$; $$f(0)=6$$; $$f(2)=20$$; $$f(4)=34$$

Exercise 1.2.50

$$f(x)= \begin{cases} x^2-2 & \text{if x < 2} \\ 4+|x-5| & \text{if x {\geq} 2} \end{cases}$$

Exercise 1.2.51

$$f(x)= \begin{cases} 5x & \text{if x < 0} \\ 3 & \text{if 0 {\geq} x {\leq} 2} \\ x^2 & \text{if x > 3} \end{cases}$$

$$f(−1)=−5$$; $$f(0)=3$$; $$f(2)=3$$; $$f(4)=16$$

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

Exercise 1.2.52

$$f(x)= \begin{cases} x+1 & \text{if x < -2} \\ -2x-3 & \text{if x {\geq} -2} \end{cases}$$

Exercise 1.2.53

$$f(x)= \begin{cases} x^2-2 & \text{if x < 1} \\ -x^2+2 & \text{if x > 1} \end{cases}$$

domain: $$(−\infty,1)\cup(1,\infty)$$

Exercise 1.2.54

$$f(x)= \begin{cases} x^2-3 & \text{if x < 0} \\ -3x^2 & \text{if x {\geq} 2} \end{cases}$$

Technology

Exercise 1.2.55

Graph $$y=\dfrac{1}{x^2}$$ on the viewing window $$[−0.5,−0.1]$$ and $$[0.1,0.5]$$. Determine the corresponding range for the viewing window. Show the graphs. window: $$[−0.5,−0.1]$$; range: $$[4, 100]$$ window: $$[0.1, 0.5]$$; range: $$[4, 100]$$

Exercise 1.2.56

Graph $$y=\dfrac{1}{x}$$ 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 $$|f(x)|$$?

$$[0, 8]$$

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 $$x>2$$.

Many answers. One function is $$f(x)=\dfrac{1}{\sqrt{x-2}}$$.

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 $$h(t)=−16t^2+96t$$. What is the domain of the function? What does the domain mean in the context of the problem?

The domain is $$[0, 6]$$; it takes 6 seconds for the projectile to leave the ground and return to the ground.
The cost in dollars of making $$x$$ items is given by the function $$C(x)=10x+500$$.
c. Suppose the maximum cost allowed is \$1500. What are the domain and range of the cost function, $$C(x)$$?