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5. Show that f(x) and g(x) are inverse functions. and g(x)=√√8x f(x) = 2²

5. Show that f(x) and g(x) are inverse functions. and g(x)=√√8x f(x) = 2²

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 5: Verify Inverse Functions** **Task:** Show that \( f(x) \) and \( g(x) \) are inverse functions. **Functions Given:** - \( f(x) = \frac{x^{3}}{8} \) - \( g(x) = \sqrt[3]{8x} \) --- **To prove that \( f(x) \) and \( g(x) \) are inverse functions, you need to demonstrate the following:** 1. \( f(g(x)) = x \) 2. \( g(f(x)) = x \) ### Solution: 1. **Compute \( f(g(x)) \):** - Substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\sqrt[3]{8x}\right) = \frac{(\sqrt[3]{8x})^3}{8} = \frac{8x}{8} = x \] 2. **Compute \( g(f(x)) \):** - Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g\left(\frac{x^{3}}{8}\right) = \sqrt[3]{8 \times \frac{x^{3}}{8}} = \sqrt[3]{x^{3}} = x \] Since both compositions \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true, \( f(x) \) and \( g(x) \) are indeed inverse functions.
Transcribed Image Text:**Problem 5: Verify Inverse Functions** **Task:** Show that \( f(x) \) and \( g(x) \) are inverse functions. **Functions Given:** - \( f(x) = \frac{x^{3}}{8} \) - \( g(x) = \sqrt[3]{8x} \) --- **To prove that \( f(x) \) and \( g(x) \) are inverse functions, you need to demonstrate the following:** 1. \( f(g(x)) = x \) 2. \( g(f(x)) = x \) ### Solution: 1. **Compute \( f(g(x)) \):** - Substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\sqrt[3]{8x}\right) = \frac{(\sqrt[3]{8x})^3}{8} = \frac{8x}{8} = x \] 2. **Compute \( g(f(x)) \):** - Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g\left(\frac{x^{3}}{8}\right) = \sqrt[3]{8 \times \frac{x^{3}}{8}} = \sqrt[3]{x^{3}} = x \] Since both compositions \( f(g(x)) = x \) and \( g(f(x)) = x \) hold true, \( f(x) \) and \( g(x) \) are indeed inverse functions.
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Step 1: First we find f^(-1)(x)

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