Answer :
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]
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]
