Example Question - finding function composition

Finding the Composition Function

The question asks you to find a function \( g(x) \) such that the composition of functions \( h(g(x)) \) is equal to the function \( f(x) \) provided. Here are the given functions: \( h(a) = \frac{1}{\sqrt{2a-1}} \) \( f(x) = x + 2 \) We want to find \( g(x) \) such that when we plug \( g(x) \) into \( h(a) \), we get \( f(x) \). So, we are solving for \( g(x) \) in the equation: \( h(g(x)) = f(x) \) \( \frac{1}{\sqrt{2g(x)-1}} = x + 2 \) Let's solve for \( g(x) \) step by step: 1. Get rid of the fraction by multiplying both sides by \( \sqrt{2g(x)-1} \): \( \sqrt{2g(x)-1} = \frac{1}{x + 2} \) 2. Square both sides to get rid of the square root: \( 2g(x)-1 = \frac{1}{(x + 2)^2} \) 3. Isolate \( g(x) \) by adding 1 to both sides: \( 2g(x) = \frac{1}{(x + 2)^2} + 1 \) 4. Divide both sides by 2 to solve for \( g(x) \): \( g(x) = \frac{1}{2(x + 2)^2} + \frac{1}{2} \) Let's simplify the equation for \( g(x) \): First, find a common denominator for the addition: \( g(x) = \frac{1 + (x + 2)^2}{2(x + 2)^2} \) Now let's expand \( (x + 2)^2 \): \( (x + 2)^2 = x^2 + 4x + 4 \) Substitute this back into \( g(x) \): \( g(x) = \frac{1 + x^2 + 4x + 4}{2(x^2 + 4x + 4)} \) This further simplifies to: \( g(x) = \frac{x^2 + 4x + 5}{2(x^2 + 4x + 4)} \) So the function \( g(x) \) that meets the requirement \( h(g(x)) = f(x) \) is: \( g(x) = \frac{x^2 + 4x + 5}{2(x^2 + 4x + 4)} \)