Deck 4: Rational, Power, and Root Functions

Question 1

(a) Sketch the graph of $f(x)=-\frac{1}{x}-2$ .
(b) Explain how the graph in part (a) is obtained from the graph of $f(x)=\frac{1}{x}$ .
(c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Answer 1

(a)

(b) The graph of $f(x)=\frac{1}{x}$ is reflected across the $x$ -axis and shifted 2 units downward.
(c)

Question 2

(a) Sketch the graph of $f(x)=-\frac{1}{x^{2}}-2$ .
(b) Explain how the graph in part (a) is obtained from the graph of $f(x)=\frac{1}{x^{2}}$ .
(c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Answer 2

(a)

(b) The graph of $f(x)=\frac{1}{x^{2}}$ is reflected across the $x$ -axis and shifted 2 units downward.
(c)

Question 3

Consider the rational function defined by $f(x)=\frac{x^{2}+3 x-4}{x^{2}-x-2}$ . Determine the answers to (a) - (e) analytically:
(a) Equations of the vertical asymptotes
(b) Equation of the horizontal asymptote
(c) $y$ -intercept
(d) $x$ -intercepts, if any
(e) Coordinates of the point where the graph of $f$ intersects its horizontal asymptote. Now sketch a comprehensive graph of $f$ .

Answer 3

(a) $x=-1 ; x=2$
(b) $y=1$
(c) 2
(d) $-4 ; 1$
(e) $\left(\frac{1}{2}, 1\right)$

Question 4

Find the equation of the oblique asymptote of the graph of the rational function defined by $f(x)=\frac{3 x^{2}-10 x+13}{x-2}$ . Then graph the function and its asymptote using a graphing calculator to illustrate an accurate comprehensive graph.

Question 5

Consider the rational function defined by $f(x)=\frac{x^{2}-16}{x-4}$ .
(a) For what value of $x$ does the graph exhibit a "hole"?
(b) Graph the function and show the "hole" in the graph.

Question 6

(a) Solve the following rational equation analytically: $\frac{3}{x-1}+\frac{4}{x^{2}-1}=\frac{13}{x+1}$ .
(b) Use the results of part (a) and a graph to find the solution set of $\frac{3}{x-1}+\frac{4}{x^{2}-1} \geq \frac{13}{x+1}$ .

Question 7

The parking attendants working at the exit of a parking ramp can process at most 15 cars per minute. If cars arrive randomly at an average rate of $x$ vehicles per minute, then the average wait $W$ in minutes for a car to exit the ramp is approximated by $W(x)=\frac{1}{15-x}$ , where $0 \leq x<15$ .
(a) Evaluate $W(10), W(14)$ , and $W(14.9)$ . Interpret the results.
(b) Graph $W$ using the window $[0,15]$ by $[-.5,1]$ . Identify the vertical asymptote. What happens to $W$ as $x$ approaches 15 ?
(c) Find $x$ when the wait is 2 minutes.

Question 8

The length and cross-sectional area of a wire determine the resistance that the wire gives to the flow of electricity. The resistance of a wire varies directly as the length of the wire and inversely as the cross-sectional area. A wire with a length of $2200 \mathrm{~cm}$ and a cross-sectional area of $5.18 \times 10^{-3} \mathrm{~cm}^{2}$ has a resistance of $.731 \mathrm{ohm}$ . If another wire has a length of $1500 \mathrm{~cm}$ and a cross-sectional area of $4.85 \times 10^{-3} \mathrm{~cm}^{2}$ , determine the resistance of the wire.

Question 9

A manufacturer needs to construct a box with a lid for a special product. The only stipulations are that the volume of the box should be 1500 cubic centimeters and that the box should have a square base. The cost for producing such a box has been determined to be represented by the function $C(x)=\frac{60+0.02 x^{3}}{x}$ , where $C(x)$ is the cost of the box in dollars and $x$ is the length of a side of the base in centimeters. Use the graph of $C$ to determine the side length $x$ that will minimize the cost of the box, and determine what this cost will be.
(Hint: Use the window $[0,50]$ by $[0,50]$ .)

Question 10

Graph the function $f(x)=\sqrt{x+3}$ in the standard viewing window. Then do each of the following:
(a) Determine the domain analytically.
(b) Use the graph to find the range.
(c) Fill in the blank with either increases or decreases: The function over its entire domain.
(d) Solve the equation $f(x)=0$ graphically.
(e) Solve the inequality $f(x)>0$ graphically.

Question 11

(a) Solve the equation $\sqrt{25-x}=x-5$ analytically. Support the solution(s) with a graph.
(b) Use the graph to find the solution set of $\sqrt{25-x}<x-5$ .
(c) Use the graph to find the solution set of $\sqrt{25-x} \geq x-5$ .

Question 12

A sidewalk in a public park is to be constructed from a restroom at point $A$ to the street at point $C$ , and another sidewalk is to be constructed from point $C$ to a swimming pool at point $B$ . Distances are shown in the figure provided. Find the distance $x$ which will minimize the combined length of the two sidewalks. What is the total length of the sidewalk to be constructed?

Question 13

(a) Sketch the graph of $f(x)=\frac{1}{x-2}+3$ .
(b) Explain how the graph in part (a) is obtained from the graph of $f(x)=\frac{1}{x}$ .
(c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Question 14

(a) Sketch the graph of $f(x)=-\frac{1}{x^{2}}+2$
(b) Explain how the graph in part (a) is obtained from the graph of $f(x)=\frac{1}{x^{2}}$ .
(c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Question 15

Consider the rational function defined by $f(x)=\frac{x^{2}-2 x-3}{x^{2}+2 x-8}$ . Determine the answers to (a) - (e) analytically:
(a) Equations of the vertical asymptotes
(b) Equation of the horizontal asymptote
(c) $y$ -intercept
(d) $x$ -intercepts, if any
(e) Coordinates of the point where the graph of $f$ intersects its horizontal asymptote. Now sketch a comprehensive graph of $f$ .

Question 16

Find the equation of the oblique asymptote of the graph of the rational function defined by $f(x)=\frac{-2 x^{2}-5 x+1}{x+1}$ . Then graph the function and its asymptote using a graphing calculator to illustrate an accurate comprehensive graph.

Question 17

Consider the rational function defined by $f(x)=\frac{x^{2}-4}{x+2}$ .
(a) For what value of $x$ does the graph exhibit a "hole"?
(b) Graph the function and show the "hole" in the graph.

Question 18

(a) Solve the following rational equation analytically: $\frac{5}{x+4}+\frac{2}{x-4}=\frac{9}{x^{2}-16}$ .
(b) Use the results of part (a) and a graph to find the solution set of $\frac{5}{x+4}+\frac{2}{x-4}<\frac{9}{x^{2}-16}$ .

Question 19

The concession stand at a sporting event can fill at most 12 orders per minute. If people arrive randomly at an average rate of $x$ people per minute, then the average wait $W$ in minutes before an order is filled is approximated by $W(x)=\frac{1}{12-x}$ where $0 \leq x<12$ .
(a) Evaluate $W(8), W(11)$ , and $W(11.9)$ . Interpret the results.
(b) Graph $W$ using the window $[0,12]$ by $[-.5,1]$ . Identify the vertical asymptote. What happens to $W$ as $x$ approaches 12 ?
(c) Find $x$ when the wait is 4 minutes.

Question 20

A building requires a beam 44 meters long, .32 meter wide, and .12 meter high. The maximum load of a horizontal beam that is supported at both ends varies directly as the width and the square of the height and inversely as the length between supports. If a beam of the same material 38 meter long, .25 meter wide, and .11 meter high can support a maximum load of 94.33 kilograms, what is the maximum load the beam in the building will support?

Question 21

A manufacturer needs to construct a box with a lid for a special product. The only stipulations are that the volume of the box should be 3000 cubic centimeters and that the box should have a square base. The cost for producing such a box has been determined to be represented by the function $C(x)=\frac{120+0.02 x^{3}}{x}$ , where $C(x)$ is the cost of the box in dollars and $x$ is the length of a side of the base in centimeters. Use the graph of $C$ to determine the side length $x$ that will minimize the cost of the box, and determine what this cost will be.
(Hint: Use the window $[0,50]$ by $[0,50]$ .)

Question 22

Graph the function $f(x)=-\sqrt{x-2}$ in the standard viewing window. Then do each of the following:
(a) Determine the domain analytically.
(b) Use the graph to find the range.
(c) Fill in the blank with either increases or decreases: The function over its entire domain.
(d) Solve the equation $f(x)=0$ graphically.
(e) Solve the inequality $f(x)<0$ graphically.

Question 23

(a) Solve the equation $\sqrt{36-x}=x-6$ analytically. Support the solution(s) with a graph.
(b) Use the graph to find the solution set of $\sqrt{36-x}<x-6$ .
(c) Use the graph to find the solution set of $\sqrt{36-x} \geq x-6$ .

Question 24

Two buildings are situated on level ground, 65 feet apart, as shown in the figure. The building on the left is 20 feet high and the other building is 40 feet high. An expensive decorative banner is to be strung from the top edge of one building to the top edge of the other, pulled tightly so that it touches the ground at point $P$ somewhere between the two buildings. Let $x$ represent the distance from $P$ to the base of the building on the left. Find the value of $x$ that will minimize the length of the banner. How long is this banner?

Question 25

(a) Sketch the graph of $f(x)=\frac{1}{x-3}+1$ .
(b) Explain how the graph in part (a) is obtained from the graph of $f(x)=\frac{1}{x}$ .
(c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Question 26

(a) Sketch the graph of $f(x)=\frac{1}{(x+2)^{2}}+1$ .
(b) Explain how the graph in part (a) is obtained from the graph of $f(x)=\frac{1}{x^{2}}$ .
(c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Question 27

Consider the rational function defined by $f(x)=\frac{x^{2}-3 x+2}{x^{2}-2 x-3}$ . Determine the answers to (a) - (e) analytically:
(a) Equations of the vertical asymptotes
(b) Equation of the horizontal asymptote
(c) $y$ -intercept
(d) $x$ -intercepts, if any
(e) Coordinates of the point where the graph of $f$ intersects its horizontal asymptote. Now sketch a comprehensive graph of $f$ .

Question 28

Find the equation of the oblique asymptote of the graph of the rational function defined by $f(x)=\frac{-3 x^{2}+7 x+4}{x-2}$ . Then graph the function and its asymptote using a graphing calculator to illustrate an accurate comprehensive graph.

Question 29

Consider the rational function defined by $f(x)=\frac{x^{2}-9}{x-3}$ .
(a) For what value of $x$ does the graph exhibit a "hole"?
(b) Graph the function and show the "hole" in the graph.

Question 30

(a) Solve the following rational equation analytically: $\frac{1}{x+2}-\frac{6}{x-2}=\frac{6}{x^{2}-4}$ .
(b) Use the results of part (a) and a graph to find the solution set of $\frac{1}{x+2}-\frac{6}{x-2} \leq \frac{6}{x^{2}-4}$ .

Question 31

The parking attendants working at the exit of a parking ramp can process at most 22 cars per minute. If cars arrive randomly at an average rate of $x$ vehicles per minute, then the average wait $W$ in minutes for a car to exit the ramp is approximated by $W(x)=\frac{1}{22-x}$ , where $0 \leq x<22$ .
(a) Evaluate $W(12), W(21)$ , and $W(21.9)$ . Interpret the results.
(b) Graph $W$ using the window $[0,22]$ by $[-.5,1]$ . Identify the vertical asymptote. What happens to $W$ as $x$ approaches 22?
(c) Find $x$ when the wait is 5 minutes.

Question 32

The length and cross-sectional area of a wire determine the resistance that the wire gives to the flow of electricity. The resistance of a wire varies directly as the length of the wire and inversely as the cross-sectional area. A wire with a length of $2400 \mathrm{~cm}$ and a cross-sectional area of $6.29 \times 10^{-3} \mathrm{~cm}^{2}$ has a resistance of $.656 \mathrm{ohm}$ . If another wire has a length of $2000 \mathrm{~cm}$ and a cross-sectional area of $7.11 \times 10^{-3} \mathrm{~cm}^{2}$ , determine the resistance of the wire.

Question 33

A manufacturer needs to construct a box with a lid for a special product. The only stipulations are that the volume of the box should be 2515 cubic centimeters and that the box should have a square base. The cost for producing such a box has been determined to be represented by the function $C(x)=\frac{100+0.02 x^{3}}{x}$ , where $C(x)$ is the cost of the box in dollars and $x$ is the length of a side of the base in centimeters. Use the graph of $C$ to determine the side length $x$ that will minimize the cost of the box, and determine what this cost will be.
(Hint: Use the window $[0,50]$ by $[0,50]$ .)

Question 34

Graph the function $f(x)=\sqrt{4-x}$ in the standard viewing window. Then do each of the following:
(a) Determine the domain analytically.
(b) Use the graph to find the range.
(c) Fill in the blank with either increases or decreases: The function over its entire domain.
(d) Solve the equation $f(x)=0$ graphically.
(e) Solve the inequality $f(x)>0$ graphically.

Question 35

(a) Solve the equation $\sqrt{16-x}=x-4$ analytically. Support the solution(s) with a graph.
(b) Use the graph to find the solution set of $\sqrt{16-x} \leq x-4$ .
(c) Use the graph to find the solution set of $\sqrt{16-x}>x-4$ .

Question 36

Two buildings are situated on level ground, 65 feet apart, as shown in the figure. The building on the left is 30 feet high and the other building is 45 feet high. An expensive decorative banner is to be strung from the top edge of one building to the top edge of the other, pulled tightly so that it touches the ground at point $P$ somewhere between the two buildings. Let $x$ represent the distance from $P$ to the base of the building on the left. Find the value of $x$ that will minimize the length of the banner. How long is this banner?

Question 37

(a) Sketch the graph of $f(x)=\frac{1}{x+3}$ .
(b) Explain how the graph in part (a) is obtained from the graph of $f(x)=\frac{1}{x}$ .
(c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Question 38

(a) Sketch the graph of $f(x)=\frac{1}{(x-1)^{2}}$ .
(b) Explain how the graph in part (a) is obtained from the graph of $f(x)=\frac{1}{x^{2}}$ .
(c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Question 39

Consider the rational function defined by $f(x)=\frac{2 x^{2}+3 x-2}{x^{2}-x-2}$ . Determine the answers to (a) - (e) analytically:
(a) Equations of the vertical asymptotes
(b) Equation of the horizontal asymptote
(c) $y$ -intercept
(d) $x$ -intercepts, if any
(e) Coordinates of the point where the graph of $f$ intersects its horizontal asymptote. Now sketch a comprehensive graph of $f$ .

Question 40

Find the equation of the oblique asymptote of the graph of the rational function defined by $f(x)=\frac{5 x^{2}-3 x-1}{x-1}$ . Then graph the function and its asymptote using a graphing calculator to illustrate an accurate comprehensive graph.

Question 41

Consider the rational function defined by $f(x)=\frac{x^{2}-25}{x+5}$ .
(a) For what value of $x$ does the graph exhibit a "hole"?
(b) Graph the function and show the "hole" in the graph.

Question 42

(a) Solve the following rational equation analytically: $\frac{2}{x^{2}-9}-\frac{3}{x+3}=\frac{7}{x-3}$ .
(b) Use the results of part (a) and a graph to find the solution set of $\frac{2}{x^{2}-9}-\frac{3}{x+3}>\frac{7}{x-3}$ .

Question 43

The concession stand at a sporting event can fill at most 9 orders per minute. If people arrive randomly at an average rate of $x$ people per minute, then the average wait $W$ in minutes before an order is filled is approximated by $W(x)=\frac{1}{9-x}$ , where $0 \leq x<9$ .
(a) Evaluate $W(6), W(8)$ , and $W(8.9)$ . Interpret the results.
(b) Graph $W$ using the window $[0,9]$ by $[-.5,1]$ . Identify the vertical asymptote. What happens to $W$ as $x$ approaches 9 ?
(c) Find $x$ when the wait is 2 minutes.

Question 44

A building requires a beam 52 meters long, .28 meter wide, and .11 meter high. The maximum load of a horizontal beam that is supported at both ends varies directly as the width and the square of the height and inversely as the length between supports. If a beam of the same material 46 meters long, .30 meter wide, and .08 meter high can support a maximum load of 49.46 kilograms, what is the maximum load the beam in the building will support?

Question 45

A manufacturer needs to construct a box with a lid for a special product. The only stipulations are that the volume of the box should be 2000 cubic centimeters and that the box should have a square base. The cost for producing such a box has been determined to be represented by the function $C(x)=\frac{80+0.02 x^{3}}{x}$ , where $C(x)$ is the cost of the box in dollars and $x$ is the length of a side of the base in centimeters. Use the graph of $C$ to determine the side length $x$ that will minimize the cost of the box, and determine what this cost will be.
(Hint: Use the window $[0,50]$ by $[0,50]$ .)

Question 46

Graph the function $f(x)=-\sqrt{5+x}$ in the standard viewing window. Then do each of the following:
(a) Determine the domain analytically.
(b) Use the graph to find the range.
(c) Fill in the blank with either increases or decreases: The function over its entire domain.
(d) Solve the equation $f(x)=0$ graphically.
(e) Solve the inequality $f(x)<0$ graphically.

Question 47

(a) Solve the equation $\sqrt{9-x}=x-3$ analytically. Support the solution(s) with a graph.
(b) Use the graph to find the solution set of $\sqrt{9-x} \leq x-3$ .
(c) Use the graph to find the solution set of $\sqrt{9-x}>x-3$ .

Question 48

A sidewalk in a public park is to be constructed from a restroom at point $A$ to the street at point $C$ , and another sidewalk is to be constructed from point $C$ to a swimming pool at point $B$ . Distances are shown in the figure provided. Find the distance $x$ which will minimize the combined length of the two sidewalks. What is the total length of the sidewalk to be constructed?