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4. Find formulas for the compositions fog and go f if f(x) = 5x² +3 and g(x) = √4x - 2. ovitettura de

4. Find formulas for the compositions fog and go f if f(x) = 5x² +3 and g(x) = √4x - 2. ovitettura de

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 4: Compositions of Functions** Given the functions \( f(x) = 5x^2 + 3 \) and \( g(x) = \sqrt{4x - 2} \), find the formulas for the compositions \( f \circ g \) and \( g \circ f \). **Solution Outline:** 1. **Composition \( f \circ g \):** - Substitute \( g(x) \) into \( f(x) \). - Calculate \( f(g(x)) = f(\sqrt{4x - 2}) \). 2. **Composition \( g \circ f \):** - Substitute \( f(x) \) into \( g(x) \). - Calculate \( g(f(x)) = g(5x^2 + 3) \). **Explanation:** - Composition \( f \circ g \) involves applying \( g(x) \) first, then applying \( f \) to the result. - Composition \( g \circ f \) involves applying \( f(x) \) first, then applying \( g \) to the result. The compositions demonstrate how combining different functions can transform inputs through a sequence of operations, illustrating the flexibility and power of mathematical functions.
Transcribed Image Text:**Problem 4: Compositions of Functions** Given the functions \( f(x) = 5x^2 + 3 \) and \( g(x) = \sqrt{4x - 2} \), find the formulas for the compositions \( f \circ g \) and \( g \circ f \). **Solution Outline:** 1. **Composition \( f \circ g \):** - Substitute \( g(x) \) into \( f(x) \). - Calculate \( f(g(x)) = f(\sqrt{4x - 2}) \). 2. **Composition \( g \circ f \):** - Substitute \( f(x) \) into \( g(x) \). - Calculate \( g(f(x)) = g(5x^2 + 3) \). **Explanation:** - Composition \( f \circ g \) involves applying \( g(x) \) first, then applying \( f \) to the result. - Composition \( g \circ f \) involves applying \( f(x) \) first, then applying \( g \) to the result. The compositions demonstrate how combining different functions can transform inputs through a sequence of operations, illustrating the flexibility and power of mathematical functions.
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