To determine the correct decomposition of [tex]\( h(x) = (g \circ f)(x) = \sqrt{x+3} \)[/tex], we need to check which pair of functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] satisfies [tex]\( h(x) \)[/tex] when composed.
Remember, [tex]\((g \circ f)(x)\)[/tex] means [tex]\( g(f(x)) \)[/tex].
Let's evaluate each option:
Option A: [tex]\( f(x) = x \)[/tex] and [tex]\( g(x) = x + 3 \)[/tex]
- When we compose these functions, we get:
[tex]\[ (g \circ f)(x) = g(f(x)) = g(x) = x + 3 \][/tex]
- However, [tex]\( h(x) = \sqrt{x+3} \neq x + 3 \)[/tex].
So, this option is not valid.
Option B: [tex]\( f(x) = \sqrt{x} \)[/tex] and [tex]\( g(x) = x + 3 \)[/tex]
- When we compose these functions, we get:
[tex]\[ (g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = \sqrt{x} + 3 \][/tex]
- [tex]\( h(x) = \sqrt{x+3} \neq \sqrt{x} + 3 \)[/tex].
Therefore, this option is also not valid.
Option C: [tex]\( f(x) = 3x \)[/tex] and [tex]\( g(x) = \sqrt{x} \)[/tex]
- When we compose these functions, we get:
[tex]\[ (g \circ f)(x) = g(f(x)) = g(3x) = \sqrt{3x} \][/tex]
- Here, [tex]\( h(x) = \sqrt{x+3} \neq \sqrt{3x} \)[/tex].
This option is not valid either.
Option D: [tex]\( f(x) = x + 3 \)[/tex] and [tex]\( g(x) = \sqrt{x} \)[/tex]
- When we compose these functions, we get:
[tex]\[ (g \circ f)(x) = g(f(x)) = g(x + 3) = \sqrt{x + 3} \][/tex]
- We observe that [tex]\( g(f(x)) = \sqrt{x + 3} = h(x) \)[/tex].
This option is valid since it satisfies [tex]\( h(x) \)[/tex] exactly.
Therefore, the correct decomposition that works for [tex]\( h(x) = \sqrt{x + 3} \)[/tex] is:
[tex]\[
\boxed{D. \; f(x) = x + 3 ; g(x) = \sqrt{x}}
\][/tex]
