To determine which expression is equivalent to [tex]\((f+g)(4)\)[/tex], let's break down the expression step-by-step:
1. Understanding the Notation:
[tex]\((f+g)(x)\)[/tex] refers to the sum of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f+g)(x) = f(x) + g(x)
\][/tex]
2. Evaluating at [tex]\(x = 4\)[/tex]:
To find [tex]\((f+g)(4)\)[/tex], we substitute [tex]\(x = 4\)[/tex]:
[tex]\[
(f+g)(4) = f(4) + g(4)
\][/tex]
3. Comparing with Given Options:
Now let’s compare [tex]\((f+g)(4) = f(4) + g(4)\)[/tex] with the given choices:
- [tex]\( f(4) + g(4) \)[/tex]: This matches exactly with our derived expression.
- [tex]\( f(x) + g(4) \)[/tex]: This expression is incorrect as it involves [tex]\(f(x)\)[/tex] instead of [tex]\(f(4)\)[/tex].
- [tex]\( f(4 + g(4)) \)[/tex]: This expression indicates that [tex]\(g(4)\)[/tex] is added to 4 inside the function [tex]\(f\)[/tex], which is not the same as evaluating [tex]\(f(4)\)[/tex] separately and then adding [tex]\(g(4)\)[/tex].
- [tex]\( 4(f(x) + g(x)) \)[/tex]: This expression implies multiplying 4 by the sum of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], which is not the same operation.
4. Conclusion:
The expression that is equivalent to [tex]\((f+g)(4)\)[/tex] is:
[tex]\[
f(4) + g(4)
\][/tex]
Therefore, the correct answer is:
[tex]\[ f(4) + g(4) \][/tex]
