To solve for [tex]\( g(x) \)[/tex] given the functions [tex]\( h(x) \)[/tex] and [tex]\( f(x) \)[/tex], we need to understand the relationship between these functions. The composition [tex]\( (f \circ g)(x) \)[/tex] means that we apply [tex]\( g(x) \)[/tex] first and then apply [tex]\( f \)[/tex].
Given the following:
[tex]\[
h(x) = \sqrt{x + 5}
\][/tex]
[tex]\[
f(x) = \sqrt{x + 2}
\][/tex]
and the relationship:
[tex]\[
h(x) = (f \circ g)(x)
\][/tex]
This implies that:
[tex]\[
h(x) = f(g(x))
\][/tex]
We can set the function definitions equal:
[tex]\[
\sqrt{x + 5} = \sqrt{g(x) + 2}
\][/tex]
To simplify this equation, we can eliminate the square roots by squaring both sides:
[tex]\[
(\sqrt{x + 5})^2 = (\sqrt{g(x) + 2})^2
\][/tex]
This simplifies to:
[tex]\[
x + 5 = g(x) + 2
\][/tex]
Now, solve for [tex]\( g(x) \)[/tex]:
[tex]\[
x + 5 = g(x) + 2 \implies g(x) = x + 5 - 2 \implies g(x) = x + 3
\][/tex]
Therefore, the function [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = x + 3
\][/tex]
