Textbook Question
Solve each rational inequality. Give the solution set in interval notation. (2x - 3)/(x + 1) > 4
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Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 59
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 asymptote by looking for the vertical line where the graph approaches infinity or negative infinity. In the graph, the red curve approaches a vertical dashed line at \(x = -2\), indicating a vertical asymptote there.
Step 2: Identify the horizontal asymptote by checking if the graph approaches a constant value as \(x\) goes to positive or negative infinity. In this graph, there is no horizontal line that the curve approaches, so there is no horizontal asymptote.
Step 3: Identify the oblique (slant) asymptote by observing if the graph approaches a non-horizontal, non-vertical line as \(x\) goes to infinity or negative infinity. The green dashed line with positive slope is the oblique asymptote, which appears to be \(y = x - 2\).
Step 4: State the domain of the function \(f\). Since there is a vertical asymptote at \(x = -2\), the function is undefined there. Therefore, the domain is all real numbers except \(x = -2\), which can be written as \((-\infty, -2) \cup (-2, \infty)\).
Step 5: Summarize the asymptotes and domain: Vertical asymptote at \(x = -2\), oblique asymptote at \(y = x - 2\), no horizontal asymptote, and domain \((-\infty, -2) \cup (-2, \infty)\).
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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, the vertical asymptote is shown as a dotted vertical line where the function spikes up or down.
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3:12
Determining Vertical Asymptotes
Horizontal and Oblique Asymptotes
Horizontal asymptotes describe the behavior of a function as the input approaches positive or negative infinity, indicating a constant value the function approaches. Oblique (slant) asymptotes occur when the function approaches a linear expression rather than a constant. In the graph, the dashed green line represents an oblique asymptote, showing the function's end behavior.
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4:48Determining Horizontal Asymptotes
Domain of a Function
The domain of a function is the set of all input values for which the function is defined. Vertical asymptotes indicate values excluded from the domain because the function is undefined there. Identifying these asymptotes helps determine the domain by excluding points where the function does not exist.
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3:51Domain Restrictions of Composed Functions
Related Practice
Textbook Question
Solve each rational inequality. Give the solution set in interval notation. (x - 8)/(x - 4) < 3
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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) = x2 + 3x + 4; k = 2+i
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Textbook Question
Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.
; no real zero less than -1
Textbook Question
Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x4+x3-x2+3; no real zero less than -2
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Textbook Question
Solve each rational inequality. Give the solution set in interval notation.