To determine which expression is equivalent to [tex]\((f+g)(4)\)[/tex], let’s carefully analyze each option and how it relates to the function notation involved.
First, let's recall that the notation [tex]\((f+g)(x)\)[/tex] means the sum of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. Specifically, [tex]\((f+g)(x) = f(x) + g(x)\)[/tex].
Given that we want to find the expression equivalent to [tex]\((f+g)(4)\)[/tex], this would mean calculating both functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex] at [tex]\(x = 4\)[/tex] and then adding their results together.
Hence, [tex]\((f+g)(4) = f(4) + g(4)\)[/tex].
Now let's examine each option:
1. [tex]\(f(4) + g(4)\)[/tex]: This directly corresponds to our expanded expression [tex]\((f+g)(4)\)[/tex].
2. [tex]\(f(x) + g(4)\)[/tex]: This is not equivalent because [tex]\(f(x)\)[/tex] involves a variable [tex]\(x\)[/tex] rather than the specific value [tex]\(4\)[/tex].
3. [tex]\(f(4 + g(4))\)[/tex]: This is not equivalent since it involves evaluating [tex]\(f\)[/tex] at a different argument [tex]\(4 + g(4)\)[/tex] rather than evaluating [tex]\(f\)[/tex] and [tex]\(g\)[/tex] separately at [tex]\(4\)[/tex].
4. [tex]\(4(f(x) + g(x))\)[/tex]: This is not equivalent because it involves multiplying the sum of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] by [tex]\(4\)[/tex], which is not the same as evaluating the sum of [tex]\(f\)[/tex] and [tex]\(g\)[/tex] at [tex]\(4\)[/tex].
Thus, the correct expression equivalent to [tex]\((f+g)(4)\)[/tex] is:
[tex]\[ f(4) + g(4) \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{f(4) + g(4)} \][/tex]
