Question

find two functions f and g such that (f ∘ g)(x) = h(x). (there are many correct answers. use non - identity functions for f(x) and g(x))
h(x) = \frac{2}{(4x + 5)^2}
(f(x), g(x)) = (\quad )
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Explanation:

Step1: Identify the inner function

We can take \( g(x) = 4x + 5 \) as the inner function.

Step2: Identify the outer function

The outer function \( f(x) \) should take the output of \( g(x) \) and produce \( h(x) \). Since \( h(x)=\frac{2}{(4x + 5)^2} \), if \( g(x)=4x + 5 \), then \( f(x)=\frac{2}{x^2} \) because when we substitute \( g(x) \) into \( f(x) \), we get \( f(g(x))=\frac{2}{(g(x))^2}=\frac{2}{(4x + 5)^2}=h(x) \).

Answer:

\( f(x)=\frac{2}{x^2} \), \( g(x)=4x + 5 \) (Note: There are other possible answers, for example, \( g(x)=4x \), \( f(x)=\frac{2}{(x + 5)^2} \) is also a valid pair of functions as long as \( (f\circ g)(x)=h(x) \))