Textbook Question
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. x/(x + 2) ≥ 2
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 60
Find the inverse of f(x) = x3 + 2
Verified step by step guidance1
Start with the function given: \(f(x) = x^3 + 2\). To find the inverse, replace \(f(x)\) with \(y\), so we have \(y = x^3 + 2\).
Swap the variables \(x\) and \(y\) to begin finding the inverse function. This gives the equation \(x = y^3 + 2\).
Solve the equation \(x = y^3 + 2\) for \(y\). Begin by isolating the cubic term: subtract 2 from both sides to get \(x - 2 = y^3\).
Next, take the cube root of both sides to solve for \(y\): \(y = \sqrt[3]{x - 2}\).
Finally, express the inverse function using function notation: \(f^{-1}(x) = \sqrt[3]{x - 2}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
An inverse function reverses the effect of the original function, meaning if f maps x to y, then its inverse maps y back to x. To find the inverse, you swap the roles of x and y and solve for y. The inverse exists only if the function is one-to-one.
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Graphing Logarithmic Functions
One-to-One Functions
A function is one-to-one if each output corresponds to exactly one input, ensuring the function passes the horizontal line test. This property is essential for the existence of an inverse function, as it guarantees that the inverse will also be a function.
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Solving Equations Involving Cubes
To find the inverse of f(x) = x^3 + 2, you need to solve for x in terms of y by isolating the cubic term and then taking the cube root. Understanding how to manipulate and solve cubic equations is crucial for expressing the inverse function explicitly.
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Related Practice
Textbook Question
Find the domain of each function.
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Textbook Question
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
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Textbook Question
Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?
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Textbook Question
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x - 2)/(x + 2) ≤ 2
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Textbook Question
Follow the seven steps to graph each rational function. f(x)=2x2/(x2−1)
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