To determine which expression is equivalent to [tex]\((f+g)(4)\)[/tex], let's first understand what the notation [tex]\((f+g)(x)\)[/tex] means.
[tex]\((f+g)(x)\)[/tex] represents the function obtained by adding the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. In other words:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Given this, we can substitute [tex]\(x = 4\)[/tex] into the expression:
[tex]\[ (f+g)(4) = f(4) + g(4) \][/tex]
Thus, the expression equivalent to [tex]\((f+g)(4)\)[/tex] is:
[tex]\[ f(4) + g(4) \][/tex]
Now let's evaluate the other given options to ensure they do not match:
1. [tex]\( f(x) + g(4) \)[/tex]:
This expression adds the value of [tex]\(f\)[/tex] at [tex]\(x\)[/tex] with the value of [tex]\(g\)[/tex] at 4, which is not the same as evaluating both functions at 4.
2. [tex]\( f(4 + g(4)) \)[/tex]:
This expression represents the function [tex]\(f\)[/tex] evaluated at the sum of 4 and [tex]\(g(4)\)[/tex], which is not the same as adding the values of [tex]\(f\)[/tex] and [tex]\(g\)[/tex] both at 4.
3. [tex]\( 4(f(x) + g(x)) \)[/tex]:
This expression takes the sum of [tex]\(f\)[/tex] and [tex]\(g\)[/tex] both evaluated at [tex]\(x\)[/tex] and multiplies that result by 4, which again is not the same as adding the values of [tex]\(f\)[/tex] and [tex]\(g\)[/tex] both at 4.
Hence, the correct and equivalent expression for [tex]\((f+g)(4)\)[/tex] is:
[tex]\[ f(4) + g(4) \][/tex]
