Question - Finding the Composition Function
Solution:
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:$$ g(x) = \frac{x^2 + 4x + 5}{2(x^2 + 4x + 4)} $$
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