To determine which expression is equivalent to [tex]\((f+g)(4)\)[/tex], let's analyze its meaning step-by-step.
1. [tex]\((f+g)(4)\)[/tex] represents the value of the composite function [tex]\((f+g)\)[/tex] evaluated at [tex]\(4\)[/tex].
2. The composite function [tex]\((f+g)(x)\)[/tex] is defined as the sum of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. Therefore, [tex]\((f+g)(x) = f(x) + g(x)\)[/tex].
3. When we evaluate this composite function at [tex]\(x = 4\)[/tex], we substitute [tex]\(4\)[/tex] in place of [tex]\(x\)[/tex]:
[tex]\[
(f+g)(4) = f(4) + g(4)
\][/tex]
4. Thus, the expression that is equivalent to [tex]\((f+g)(4)\)[/tex] is [tex]\(f(4) + g(4)\)[/tex].
Based on the given choices:
- [tex]\(f(4) + g(4)\)[/tex] correctly represents [tex]\((f+g)(4)\)[/tex].
- [tex]\(f(x) + g(4)\)[/tex] mixes the function evaluated at [tex]\(4\)[/tex] and [tex]\(x\)[/tex], which is not correct.
- [tex]\(f(4 + g(4))\)[/tex] suggests a function within a function which is not applicable here.
- [tex]\(4(f(x) + g(x))\)[/tex] incorrectly scales the sum of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] by [tex]\(4\)[/tex], which is not equivalent.
Therefore, the equivalent expression to [tex]\((f+g)(4)\)[/tex] is:
[tex]\[
f(4) + g(4)
\][/tex]
