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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 17
Chapter 3, Problem 17

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) = x³ +2

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1
Start with the given function: \(f(x) = x^3 + 2\). To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = x^3 + 2\).
Swap the variables \(x\) and \(y\) to begin solving for the inverse: \(x = y^3 + 2\).
Isolate the cubic term by subtracting 2 from both sides: \(x - 2 = y^3\).
Take the cube root of both sides to solve for \(y\): \(y = \sqrt[3]{x - 2}\). This expression represents the inverse function \(f^{-1}(x)\).
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 and vice versa, ensuring that no two different inputs produce the same output. This property is essential for a function to have an inverse, as the inverse must also be a function.
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Inverse Functions

The inverse of a function reverses the roles of inputs and outputs, meaning if f(x) maps x to y, then f⁻¹(y) maps y back to x. Finding the inverse involves solving the original function's equation for x in terms of y.
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Verification of Inverse Functions

To verify that two functions are inverses, you must show that composing them in either order returns the original input: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This confirms the functions undo each other's operations.
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Related Practice
Textbook Question

In Exercises 11–26, determine whether each equation defines y as a function of x. x = y²

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Textbook Question

Use the graph to determine (a) the function's domain, (b) the function's range, (c) the x-intercepts, if any, (d) the y-intercept, if there is one, (e) intervals on which the function is increasing, decreasing or constant, (f) the missing function values, indicated by question marks, below each graph.

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Textbook Question

Find the domain of each function. f(x) = √(x - 3)

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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

Find the average rate of change of the function from x1 to x2. f(x) = √x from x1 = 4 to x2 = 9

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Textbook Question

Find the distance between each pair of points. If necessary, express answers in simplified radical form and then round to two decimal places. (7/3, 1/5) and (1/3, 6/5)

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