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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.1.25
Chapter 7, Problem 7.1.25

Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x⁵

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Start with the given function: \(f(x) = x^5\). Our goal is to find the inverse function \(f^{-1}(x)\), which means we want to express \(x\) in terms of \(y\) where \(y = f(x)\).
Replace \(f(x)\) with \(y\): \(y = x^5\). To find the inverse, solve this equation for \(x\) in terms of \(y\).
Take the fifth root of both sides to isolate \(x\): \(x = \sqrt[5]{y} = y^{1/5}\). This gives the inverse function: \(f^{-1}(x) = x^{1/5}\).
Determine the domain and range of \(f^{-1}(x)\). Since \(f(x) = x^5\) is defined for all real numbers and is one-to-one, its inverse will also have domain and range as all real numbers.
Verify the inverse by checking the compositions: compute \(f(f^{-1}(x)) = (x^{1/5})^5\) and \(f^{-1}(f(x)) = (x^5)^{1/5}\), and confirm both simplify to \(x\).

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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, swapping inputs and outputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Finding the inverse involves solving y = f(x) for x in terms of y.
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Domain and Range of Functions and Their Inverses

The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs. For inverse functions, the domain and range swap roles: the domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
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Properties of Power Functions

Power functions like f(x) = x⁵ are continuous and one-to-one over all real numbers, making them invertible. Since x⁵ is strictly increasing, its inverse is the fifth root function, which also has domain and range as all real numbers.
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