To express the function [tex]\( h(x) = \frac{4}{3 - \sqrt{4 + x^2}} \)[/tex] as the composition of two functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] in such a way that [tex]\( h(x) = (f \circ g)(x) = f(g(x)) \)[/tex], we need to find appropriate functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex].
Let's start by identifying the inner and outer functions corresponding to the given form [tex]\( h(x) \)[/tex].
Given [tex]\( h(x) = \frac{4}{3 - \sqrt{4 + x^2}} \)[/tex]:
1. It's evident that the inner part involves [tex]\( \sqrt{4 + x^2} \)[/tex]. Let’s define [tex]\( g(x) \)[/tex] such that:
[tex]\[ g(x) = \sqrt{4 + x^2} \][/tex]
This simplifies the expression [tex]\( h(x) \)[/tex] to:
[tex]\[ h(x) = \frac{4}{3 - g(x)} \][/tex]
2. Next, we need to determine [tex]\( f(x) \)[/tex] such that [tex]\( f(g(x)) = h(x) \)[/tex]. Here [tex]\( g(x) = \sqrt{4 + x^2} \)[/tex] is already defined. So we get:
[tex]\[ f(g(x)) = \frac{4}{3 - g(x)} \][/tex]
Therefore, our function [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = \frac{4}{3 - x} \][/tex]
Putting these functions together, we have the composition of the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] such that:
[tex]\[ (f \circ g)(x) = f(g(x)) = \frac{4}{3 - \sqrt{4 + x^2}} \][/tex]
Thus, the correct answer is:
[tex]\[ f(x) = \frac{4}{3 - x} \][/tex]
[tex]\[ g(x) = \sqrt{4 + x^2} \][/tex]
