Textbook Question
For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)2(x-5)2
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 55
Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.
Verified step by step guidance1
Step 1: Identify the vertical asymptotes by looking for values of x where the function approaches infinity or negative infinity. In the graph, these are the vertical dashed lines where the function is undefined. Here, the vertical asymptotes are at \(x = -4\) (red dashed line) and \(x = 3\) (orange dashed line).
Step 2: Identify the horizontal asymptote by observing the behavior of the function as \(x\) approaches positive or negative infinity. The graph shows the function approaching the horizontal dashed purple line \(y = -2\) as \(x\) goes to both \(+\infty\) and \(-\infty\). So, the horizontal asymptote is \(y = -2\).
Step 3: Check for any oblique (slant) asymptotes by looking for a line that the graph approaches as \(x\) goes to infinity, which is neither horizontal nor vertical. In this graph, there is no oblique asymptote since the function approaches a horizontal line instead.
Step 4: State the domain of the function \(f\). The domain includes all real numbers except where the vertical asymptotes occur, because the function is undefined at those points. Therefore, the domain is all real numbers except \(x = -4\) and \(x = 3\).
Step 5: Summarize the asymptotes and domain: Vertical asymptotes at \(x = -4\) and \(x = 3\), horizontal asymptote at \(y = -2\), no oblique asymptote, and domain \((-\infty, -4) \cup (-4, 3) \cup (3, \infty)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertical Asymptotes
Vertical asymptotes occur where the function approaches infinity or negative infinity as the input approaches a specific value. These are typically found where the denominator of a rational function is zero, causing the function to be undefined. In the graph, vertical asymptotes are shown as vertical dashed lines where the curve shoots up or down without bound.
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Determining Vertical Asymptotes
Horizontal and Oblique Asymptotes
Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity, indicating the value the function approaches. Oblique (slant) asymptotes occur when the function approaches a line with a non-zero slope at infinity. In the graph, the horizontal asymptote is a horizontal dashed line, while an oblique asymptote would be a slanted dashed line.
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4:48Determining Horizontal Asymptotes
Domain of a Function
The domain of a function is the set of all input values (x-values) for which the function is defined. For rational functions, the domain excludes values that cause division by zero, often corresponding to vertical asymptotes. Identifying vertical asymptotes helps determine the domain by excluding those x-values.
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3:51Domain Restrictions of Composed Functions
Related Practice
Textbook Question
Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zeros of -2, 1, and 0; ƒ(-1)=-1
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Textbook Question
For each polynomial function, identify its graph from choices A–F.
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Textbook Question
What happens to y if y varies directly as x, and x is halved?
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Textbook Question
Solve each rational inequality. Give the solution set in interval notation. (x + 1)/(x - 5) > 0
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Textbook Question
Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = 5x4 + 2x3 -x+3; k=2/5