Exam 4: Rational, Power, and Root Functions

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.

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Question 1

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$(a) [-3, \infty) (b) [0, \infty) (c) increases (d) \{-3\} (e) (-3, \infty)$ (a) $[-3, \infty)$
(b) $[0, \infty)$
(c) increases
(d) $\{-3\}$
(e) $(-3, \infty)$

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.

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Question 2

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(a) -2
(b)

(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).

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Question 3

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

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.

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Question 4

(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 $\frac{2}{x^{2}-9}-\frac{3}{x+3}>\frac{7}{x-3}$ .

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Question 5

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 $f$ .

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Question 6

(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 $\sqrt{25-x} \geq x-5$ .

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Question 7

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.

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Question 8

(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).

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Question 9

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.

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Question 10

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 $f$ .

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Question 11

(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 $\sqrt{36-x} \geq x-6$ .

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Question 12

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 $f$ .

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Question 13

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 $[0,50]$ .)

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Question 14

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.

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Question 15

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 $[0,50]$ .)

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Question 16

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.

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Question 17

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.

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Question 18

(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).

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Question 19

(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).

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Question 20

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.

(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).

(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 $\frac{2}{x^{2}-9}-\frac{3}{x+3}>\frac{7}{x-3}$ .

(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 $\sqrt{25-x} \geq x-5$ .

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.

(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).

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.

(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 $\sqrt{36-x} \geq x-6$ .

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.

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.

(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).

(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).

Linear Functions, Equations, and Inequalities

Analysis of Graphs of Functions

Polynomial Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Analytic Geometry and Nonlinear Systems

The Unit Circle and Functions of Trigonometry

Trigonometric Identities and Equations

Applications of Trigonometry and Vectors

Further Topics in Algebra

Limits, Derivatives, and Definite Integrals

Reference: Basic Algebraic Concepts

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Exam 4: Rational, Power, and Root Functions

Consider the rational function defined by f(x)=x2−4x+2f(x)=\frac{x^{2}-4}{x+2}f(x)=x+2x2−4​ . (a) For what value of xxx does the graph exhibit a "hole"? (b) Graph the function and show the "hole" in the graph.

(a) Sketch the graph of f(x)=−1x2−2f(x)=-\frac{1}{x^{2}}-2f(x)=−x21​−2 . (b) Explain how the graph in part (a) is obtained from the graph of f(x)=1x2f(x)=\frac{1}{x^{2}}f(x)=x21​ . (c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Find the equation of the oblique asymptote of the graph of the rational function defined by f(x)=−3x2+7x+4x−2f(x)=\frac{-3 x^{2}+7 x+4}{x-2}f(x)=x−2−3x2+7x+4​ . Then graph the function and its asymptote using a graphing calculator to illustrate an accurate comprehensive graph.

(a) Sketch the graph of f(x)=−1x−2f(x)=-\frac{1}{x}-2f(x)=−x1​−2 . (b) Explain how the graph in part (a) is obtained from the graph of f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ . (c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Find the equation of the oblique asymptote of the graph of the rational function defined by f(x)=3x2−10x+13x−2f(x)=\frac{3 x^{2}-10 x+13}{x-2}f(x)=x−23x2−10x+13​ . Then graph the function and its asymptote using a graphing calculator to illustrate an accurate comprehensive graph.

Find the equation of the oblique asymptote of the graph of the rational function defined by f(x)=5x2−3x−1x−1f(x)=\frac{5 x^{2}-3 x-1}{x-1}f(x)=x−15x2−3x−1​ . Then graph the function and its asymptote using a graphing calculator to illustrate an accurate comprehensive graph.

Consider the rational function defined by f(x)=x2−9x−3f(x)=\frac{x^{2}-9}{x-3}f(x)=x−3x2−9​ . (a) For what value of xxx does the graph exhibit a "hole"? (b) Graph the function and show the "hole" in the graph.

(a) Sketch the graph of f(x)=1x+3f(x)=\frac{1}{x+3}f(x)=x+31​ . (b) Explain how the graph in part (a) is obtained from the graph of f(x)=1xf(x)=\frac{1}{x}f(x)=x1​ . (c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

(a) Sketch the graph of f(x)=1(x−1)2f(x)=\frac{1}{(x-1)^{2}}f(x)=(x−1)21​ . (b) Explain how the graph in part (a) is obtained from the graph of f(x)=1x2f(x)=\frac{1}{x^{2}}f(x)=x21​ . (c) Use a graphing calculator to obtain an accurate depiction of the graph in part (a).

Linear Functions, Equations, and Inequalities

Analysis of Graphs of Functions

Polynomial Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Analytic Geometry and Nonlinear Systems

The Unit Circle and Functions of Trigonometry

Trigonometric Identities and Equations

Applications of Trigonometry and Vectors

Further Topics in Algebra

Limits, Derivatives, and Definite Integrals

Reference: Basic Algebraic Concepts

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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.

(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).

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.

(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).

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.

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.

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.

(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).

(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).