Textbook Question
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).
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Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 51
Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. ƒ(x)=2x4-4x2+4x-8; 1 and 2
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Recall the Intermediate Value Theorem (IVT), which states that if a function \( f(x) \) is continuous on a closed interval \([a, b]\) and \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one \( c \) in \((a, b)\) such that \( f(c) = 0 \).
Identify the function \( f(x) = 2x^4 - 4x^2 + 4x - 8 \) and the interval \([1, 2]\). Since \( f(x) \) is a polynomial, it is continuous everywhere, including on \([1, 2]\).
Evaluate \( f(1) \) by substituting \( x = 1 \) into the function: \( f(1) = 2(1)^4 - 4(1)^2 + 4(1) - 8 \).
Evaluate \( f(2) \) by substituting \( x = 2 \) into the function: \( f(2) = 2(2)^4 - 4(2)^2 + 4(2) - 8 \).
Check the signs of \( f(1) \) and \( f(2) \). If one is positive and the other is negative, then by the IVT, there is at least one real zero \( c \) in the interval \((1, 2)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Intermediate Value Theorem
The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes values f(a) and f(b) at each end, then it must take any value between f(a) and f(b) at some point within the interval. This theorem is used to prove the existence of roots when the function changes sign.
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Polynomial Continuity
Polynomial functions are continuous everywhere on the real number line, meaning there are no breaks, jumps, or holes in their graphs. This continuity ensures that the Intermediate Value Theorem can be applied to any interval when analyzing polynomial functions.
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06:04Introduction to Polynomial Functions
Evaluating Function Values to Detect Sign Change
To apply the Intermediate Value Theorem, you evaluate the polynomial at the endpoints of the interval. If the function values at these points have opposite signs (one positive, one negative), it guarantees at least one real zero exists between them, indicating the function crosses the x-axis.
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4:26Evaluating Composed Functions
Related Practice
Textbook Question
For each polynomial function, find all zeros and their multiplicities.
Textbook Question
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).
Textbook Question
Work each problem. Which function has a graph that does not have a vertical asymptote?
A. ƒ(x)=1/(x2+2)
B. ƒ(x)=1/(x2-2)
C. ƒ(x)=3/x2
D. ƒ(x)=(2x+1)/(x-8)
Textbook Question
Connecting Graphs with Equations Find a quadratic function f having the graph shown. (Hint: See the Note following Example 3.)
Textbook Question
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).
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