Textbook Question
In Exercises 11–26, determine whether each equation defines y as a function of x. 4x = y²
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Ch. 2 - Functions and Graphs
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 3, Problem 19
The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ-1 (x)) = = x and ƒ-1 (f(x)) = x. f(x) = (x+2)³
Verified step by step guidance1
Start with the given function: \(f(x) = (x + 2)^3\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = (x + 2)^3\).
Swap the variables \(x\) and \(y\) to begin finding the inverse: \(x = (y + 2)^3\).
Solve the equation for \(y\) by taking the cube root of both sides: \(\sqrt[3]{x} = y + 2\).
Isolate \(y\) by subtracting 2 from both sides: \(y = \sqrt[3]{x} - 2\). This expression represents the inverse function, so write \(f^{-1}(x) = \sqrt[3]{x} - 2\).
To verify the inverse, compute \(f(f^{-1}(x))\) by substituting \(f^{-1}(x)\) into \(f(x)\) and simplify to check if it equals \(x\). Then compute \(f^{-1}(f(x))\) by substituting \(f(x)\) into \(f^{-1}(x)\) and simplify to check if it equals \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
One-to-One Functions
A one-to-one function assigns each input a unique output, ensuring no two different inputs share the same output. This property is essential for a function to have an inverse, as the inverse must reverse the mapping uniquely.
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Decomposition of Functions
Inverse Functions
The inverse of a function f, denoted f⁻¹, reverses the effect of f, swapping inputs and outputs. To find f⁻¹(x), solve the equation y = f(x) for x in terms of y, then interchange variables. The inverse exists only if f is one-to-one.
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Verification of Inverse Functions
To verify that two functions are inverses, show that composing them returns the original input: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This confirms that each function undoes the action of the other, proving they are true inverses.
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Related Practice
Textbook Question
use the graph of y = f(x) to graph each function g.
g(x) = f(x-1)
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Textbook Question
Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ passes through (−1, 5) and is perpendicular to the line whose equation is x = 6.
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Textbook Question
Find the midpoint of each line segment with the given endpoints. (6, 8) and (2, 4)
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Textbook Question
Determine whether each equation defines y as a function of x. y = √x +4
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Textbook Question
Find the domain of each function. f(x) = 1/√(x - 3)
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