To determine which expression is equivalent to [tex]\((f+g)(4)\)[/tex], let's break it down step-by-step.
1. Understanding [tex]\((f+g)(4)\)[/tex]:
- The notation [tex]\((f+g)(x)\)[/tex] represents the sum of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] evaluated at [tex]\(x\)[/tex].
- Therefore, [tex]\((f+g)(4)\)[/tex] means we need to evaluate both [tex]\(f\)[/tex] and [tex]\(g\)[/tex] at [tex]\(x = 4\)[/tex] and then add their results.
2. Evaluating [tex]\((f+g)(4)\)[/tex]:
- By definition, [tex]\((f+g)(x) = f(x) + g(x)\)[/tex].
- Applying this to the specific case [tex]\(x = 4\)[/tex], we get:
[tex]\[
(f+g)(4) = f(4) + g(4)
\][/tex]
3. Comparison with given options:
- [tex]\(f(4) + g(4)\)[/tex] is indeed the sum of [tex]\(f(4)\)[/tex] and [tex]\(g(4)\)[/tex], matching our derived expression.
- [tex]\(f(x) + g(4)\)[/tex] mixes variables, which does not match our exact requirement since we only consider [tex]\(x = 4\)[/tex].
- [tex]\(f(4+g(4))\)[/tex] involves evaluating [tex]\(f\)[/tex] at [tex]\(4 + g(4)\)[/tex], which is not the same as simply adding [tex]\(f(4)\)[/tex] and [tex]\(g(4)\)[/tex].
- [tex]\(4(f(x) + g(x))\)[/tex] scales the sum of the functions by 4 for any [tex]\(x\)[/tex], which is unrelated to evaluating each function at [tex]\(x = 4\)[/tex] separately and then summing.
Thus, the correct equivalent expression to [tex]\((f+g)(4)\)[/tex] is:
[tex]\[ f(4) + g(4) \][/tex]
This matches our understanding and evaluation.
