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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 13
Chapter 3, Problem 13

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) = 2x

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Start with the given function: \(f(x) = 2x\). To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\), so you have \(y = 2x\).
Next, interchange the variables \(x\) and \(y\) to reflect the inverse relationship. This gives \(x = 2y\).
Now, solve this equation for \(y\) to express the inverse function. Divide both sides by 2 to isolate \(y\): \(y = \frac{x}{2}\).
Rewrite \(y\) as \(f^{-1}(x)\) to get the inverse function: \(f^{-1}(x) = \frac{x}{2}\).
To verify the inverse, compute \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\). Substitute \(f^{-1}(x)\) into \(f(x)\) and simplify, then substitute \(f(x)\) into \(f^{-1}(x)\) and simplify. Both should simplify to \(x\) if the inverse is correct.

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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, effectively 'undoing' the original function. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, meaning applying one after the other returns the original value.
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Verification of Inverse Functions

To verify that two functions are inverses, you must show that composing them in both orders returns the input variable. Specifically, f(f⁻¹(x)) = x and f⁻¹(f(x)) = x must hold true for all x in the domains of the respective functions.
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Related Practice
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Use the graph of y = f(x) to graph each function g.

g(x) = f(x/2)

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Textbook Question

In Exercises 11–26, determine whether each equation defines y as a function of x. x² + y = 25