1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

4. Polynomial and Rational Functions

5. Exponential and Logarithmic Functions

6. Systems of Equations and Inequalities

7. Matrices and Determinants

8. Conic Sections

9. Sequences, Induction, and Probability

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In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. g(x)=1/(x−1)

Zeros and Vertical Asymptotes of a Rational Function

Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x^2? What is its range?

Rational Functions of the Form y = 1/x^n and Their Transformations / Example 11.5

In Exercises 1–8, find the domain of each rational function. g(x)=3x^2/(x−5)(x+4)

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(x^2-2x-3)/(2x^2-x-10)

Find the Equations of the Vertical and Horizontal Asymptotes of Rational Function f(x)=(3x+12)/(x+2)

Use the graphs of the rational functions in choices A–D to answer each question. There may be more than one correct choice.  Which choices have domain (-∞, 3)U(3, ∞)?

In Exercises 95–98, use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function's equation and transformations of f(x) = 1/x to graph g. g(x) = (2x+7)/(x+3)

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/3a9bbc44400e2587-1683810816284.jpg" alt="" /> As x -> ∞, f(x) -> __

Graph each rational function. See Examples 5–9. ƒ(x)=(9x^2-1)/(x^2-4)

Finding All Asymptotes of a Rational Function (Vertical, Horizontal, Oblique / Slant)

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=4x/(x−2)

 General Attributes of Rational Functions

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/25b925f2c7f1c083-1675372297034.jpg" alt="" /> As x -> -3^-, f(x) -> __

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=(x+7)/(x^2+4x−21)

Solve each problem. Find a rational function ƒ having the graph shown. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/067fddca3f8aa514-1678790615999.jpg" alt="" />

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d). 

ƒ(x)=-1/(x-2)^2

Rational Equations- Work-rate Problem

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^2+x−6)/(x−3)

Arrow Notation

Comprehensive Graph of a Rational Function

Graph each rational function. ƒ(x)=(4x-2)/(3x+1)

Rational Functions: Vertical and Horizontal Asymptotes, Arrow notation

Match the rational function in Column I with the appropriate descrip-tion in Column II. Choices in Column II can be used only once. ƒ(x)=(x^2-16)/(x+4)

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/0d03926f45ada28f-1675372401938.jpg" alt="" /> As x -> -∞, f(x) -> __

Provide a short answer to each question. What is the equation of the vertical asymptote of the graph of y=1/(x-3)+2? Of the horizontal asymptote?

Graph each rational function. See Examples 5–9. ƒ(x)=3x/(x^2-x-2)

Coordinates of a Hole of a Rational Function

Arrow Notation -Rational Functions

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d).

ƒ(x)=1/(x-2)^2

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. r(x) = (x^2 + 4x + 3)/(x + 2)^2

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^3+1)/(x^2+2x)

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=(x^2+4x−21)/(x+7)

Rational Function : Find the Asymptotes (Vertical, Horizontal, Slant / Oblique)

 Rational Functions of the Form y = 1/x^n and Their Transformations / Example 11.6

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/eb0658cf12f3ac47-1679047994479.jpg" alt="" />

Finding Vertical Asymptotes of Rational Functions

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^2+1)/x

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/30b028a1f5e63bd0-1679049927699.jpg" alt="" />

Graph Rational Functions

Rational Functions and Their Graphs

Graphing a Rational Function that has an Oblique/Slant Asymptote and  a Vertical Asymptote

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(4x^2+25)/(x^2+9)

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d).

ƒ(x)=-1/(x-2)

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(3x^2+x−4)/(2x^2−5x)

Work each problem. Which function has a graph that does not have a vertical asymptote?

A. ƒ(x)=1/(x^2+2) B. ƒ(x)=1/(x^2-2) C. ƒ(x)=3/x^2 D. ƒ(x)=(2x+1)/(x-8)

Graphing a Rational Function with Slant / Oblique Asymptote

Vertical Asymptotes of Rational Functions: Quick Way to Find Them, Another Example 2

How to Find Slant Asymptote of a Rational Function

Graph each rational function. ƒ(x)=4/(x-1)

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=2/(x^2+x−2)

Graph each rational function. See Examples 5–9. ƒ(x)=(x+1)/(x-4)

 Finding Horizontal Asymptotes Using Leading Terms

Graph each rational function. See Examples 5–9. ƒ(x)=(16x^2-9)/(x^2-9)

In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. (1 − 3/(x+2)) / (1 + 1/(x−2))

Identify a vertical asymptote

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(4-3x)/(2x+1)

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/ae00fe439f546036-1679048800379.jpg" alt="" />

Match the rational function in Column I with the appropriate descrip-tion in Column II. Choices in Column II can be used only once. ƒ(x)=(x+7)/(x+1)

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x^3/(3x^2+1)

Solve each problem. Work each of the following. Sketch the graph of a function that does not intersect its horizontal asymptote y=1, has the line x=3 as a vertical asymptote, and has x-intercepts (2, 0) and (4, 0).

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/ff21d62ae02a27ee-1675372430629.jpg" alt="" /> As x -> 1^-, f(x) -> __

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. f(x)=x/(x+4)

Finding Oblique Asymptotes by Long Division

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=x/(x^2+4)

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. h(x)=1/x + 2

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^2−1)/x

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. g(x)=1/(x+1) − 2

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. f(x) = 2x/(x^2 - 9)

Graphing Rational Functions and Their Asymptotes

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. f(x)=(x^2−9)/(x−3)

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/02db0cd75ab33e1d-1675451421012.jpg" alt="" /> As x -> 1^+, f(x) -> __

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(x^2+1)/(x^2+9)

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(x^2+4)/(x-1)

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (4x^2 - 16x + 16)/(2x - 3)

Find a rational function ƒ having a graph with the given features.
x-intercepts: (1, 0) and (3, 0)
y-intercept: none 
vertical asymptotes: x=0 and x=2
horizontal asymptote: y=1

Shortcut to Find Horizontal Asymptotes of Rational Functions

In Exercises 1–8, find the domain of each rational function. f(x)=5x/(x−4)

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=x^4/(x^2+2)

Graph each rational function. See Examples 5–9. ƒ(x)=(2x+1)/(x^2+6x+8)

Solve each problem. Work each of the following. Find an equation for a possible corresponding rational function.

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. g(x)=(x−3)/(x^2−9)

Provide a short answer to each question. What is the domain of the function ƒ(x)=1/x? What is its range?

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=2x^2/(x^2+4)

Create and graph relationships using rational functions

Find a rational function ƒ having a graph with the given features. 
x-intercepts: (-1, 0) and (3, 0)
y-intercept: (0, -3)
vertical asymptote: x=1
horizontal asymptote: y=1

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(x^2-1)/(x+3)

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=(−2x+1)/(3x+5)

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. g(x)=1/(x+2)^2

In Exercises 55–56, use transformations of f(x) = (1/x) or f(x) = (1/x^2) to graph each rational function. g(x) = 1/(x + 2)^2 - 1<img src="https://lightcat-files.s3.amazonaws.com/problem_images/e42381fd18f9909d-1672078344441.jpg" alt="" />

Rational Equations- Applications with Uniform Motion

Provide a short answer to each question. Is ƒ(x)=1/x^2 an even or an odd function? What symmetry does its graph exhibit?

Match the rational function in Column I with the appropriate descrip-tion in Column II. Choices in Column II can be used only once. ƒ(x)=1/(x+4)

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. g(x)=(x+3)/x(x+4)

MATHS - Graphs - Asymptotes

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. h(x)=1/(x−3)^2+1

Finding the Domain of Rational Functions

In Exercises 45–56, use transformations of f(x)=1/x or f(x)=1/x^2 to graph each rational function. h(x)=1/x2 − 4

Using Arrow Notation

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(2x+6)/(x-4)

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x−2)/(x^2−4)

In Exercises 1–8, find the domain of each rational function. h(x)=(x+7)/(x^2−49)

In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=x/x(x+4)

 Rational Functions of the Form y = 1/x^n and Their Transformations / Example 11.4

Match the rational function in Column I with the appropriate descrip-tion in Column II. Choices in Column II can be used only once. ƒ(x)=(x^2+3x+4)/(x-5)

Provide a short answer to each question. Is ƒ(x)=1/x an even or an odd function? What symmetry does its graph exhibit?

Vertical Asymptotes of Rational Functions: Quick Way to Find Them, Another Example 1

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=12x/(3x^2+1)

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=3/(x-5)

In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x+2)/(x^2+x−6)

How to find SLANT ASYMPTOTES 

Work each problem. Which function has a graph that does not have a horizontal asymptote?

A. ƒ(x)=(2x-7)/(x+3) B. ƒ(x)=3x/(x^2-9) 
C. ƒ(x)=(x^2-9)/(x+3) D. ƒ(x)=(x+5)/(x+2)(x-3)

Finding the Domain of a Rational Function / Fraction with Variable in Denominator

In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. x/(2x+6) − 9/(x^2−9)

Rational Expressions and Domain

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/316e217bab49da77-1675452034423.jpg" alt="" /> As x -> -2^+, f(x) -> __

In Exercises 1–8, find the domain of each rational function. f(x)=(x+7)/(x^2+49)

Finding the Domain of an Expression Involving Fractions - Example 1

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ. <img src="https://lightcat-files.s3.amazonaws.com/problem_images/612c3806f20d7479-1680110748044.jpg" alt="" />

In Exercises 37–44, find the horizontal asymptote, if there is one, of the graph of each rational function. g(x)=12x^2/(3x^2+1)

In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. 5x^2/(x2−4) ⋅ (x^2+4x+4)/(10x^3)

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d). 

ƒ(x)=1/(x-2)

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