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Determine whether the two functions g(x) = (r - 5)3 and h(x) = Vr+5 %3D %3D are inverses.

Determine whether the two functions g(x) = (r - 5)3 and h(x) = Vr+5 %3D %3D are inverses.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Determine whether the two functions \( g(x) = (x - 5)^3 \) and \( h(x) = \sqrt[3]{x} + 5 \) are inverses.** ### Explanation To determine if two functions are inverses, you can verify if their composition results in the identity function. Specifically, if \( g(h(x)) = x \) and \( h(g(x)) = x \) for all \( x \) in the domains, then the functions are inverses of each other. #### Step-by-Step Guide 1. **Compose \( g(h(x)) \):** - Substitute \( h(x) \) into \( g(x) \): \( g(h(x)) = g(\sqrt[3]{x} + 5) \). - Simplify: \( g(\sqrt[3]{x} + 5) = (\sqrt[3]{x} + 5 - 5)^3 = (\sqrt[3]{x})^3 = x \). 2. **Compose \( h(g(x)) \):** - Substitute \( g(x) \) into \( h(x) \): \( h(g(x)) = h((x - 5)^3) \). - Simplify: \( h((x - 5)^3) = \sqrt[3]{(x - 5)^3} + 5 = x - 5 + 5 = x \). Since both compositions \( g(h(x)) \) and \( h(g(x)) \) simplify to \( x \), the functions \( g(x) \) and \( h(x) \) are indeed inverses.
Transcribed Image Text:**Determine whether the two functions \( g(x) = (x - 5)^3 \) and \( h(x) = \sqrt[3]{x} + 5 \) are inverses.** ### Explanation To determine if two functions are inverses, you can verify if their composition results in the identity function. Specifically, if \( g(h(x)) = x \) and \( h(g(x)) = x \) for all \( x \) in the domains, then the functions are inverses of each other. #### Step-by-Step Guide 1. **Compose \( g(h(x)) \):** - Substitute \( h(x) \) into \( g(x) \): \( g(h(x)) = g(\sqrt[3]{x} + 5) \). - Simplify: \( g(\sqrt[3]{x} + 5) = (\sqrt[3]{x} + 5 - 5)^3 = (\sqrt[3]{x})^3 = x \). 2. **Compose \( h(g(x)) \):** - Substitute \( g(x) \) into \( h(x) \): \( h(g(x)) = h((x - 5)^3) \). - Simplify: \( h((x - 5)^3) = \sqrt[3]{(x - 5)^3} + 5 = x - 5 + 5 = x \). Since both compositions \( g(h(x)) \) and \( h(g(x)) \) simplify to \( x \), the functions \( g(x) \) and \( h(x) \) are indeed inverses.
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