Answer :
To find an expression equivalent to [tex]\((f + g)(4)\)[/tex], you need to understand the composition of functions. When we denote [tex]\((f + g)(4)\)[/tex], it means we first evaluate each function at [tex]\(x = 4\)[/tex] and then add those results.
Let's break it down step-by-step:
1. Evaluate each function at [tex]\(x = 4\)[/tex]:
- This means we need the value of [tex]\(f(4)\)[/tex] and the value of [tex]\(g(4)\)[/tex].
2. Add the results:
- After finding [tex]\(f(4)\)[/tex] and [tex]\(g(4)\)[/tex], sum these two values.
Thus, [tex]\((f + g)(4)\)[/tex] can be written as:
[tex]\[ (f + g)(4) = f(4) + g(4) \][/tex]
Examining the given choices:
1. [tex]\(f(4) + g(4)\)[/tex] - This directly follows from our breakdown and is the correct expression.
2. [tex]\(f(x) + g(4)\)[/tex] - This is incorrect because it involves a function value dependent on [tex]\(x\)[/tex] and another fixed at 4.
3. [tex]\(f(4 + g(4))\)[/tex] - This is incorrect because it implies substituting 4 + g(4) into the function [tex]\(f\)[/tex], which is not what we need.
4. [tex]\(4(f(x) + g(x))\)[/tex] - This is incorrect as it scales the sum of functions by 4 and applies to all [tex]\(x\)[/tex], not just 4.
Therefore, the equivalent expression to [tex]\((f + g)(4)\)[/tex] is:
[tex]\[ f(4) + g(4) \][/tex]
Let's break it down step-by-step:
1. Evaluate each function at [tex]\(x = 4\)[/tex]:
- This means we need the value of [tex]\(f(4)\)[/tex] and the value of [tex]\(g(4)\)[/tex].
2. Add the results:
- After finding [tex]\(f(4)\)[/tex] and [tex]\(g(4)\)[/tex], sum these two values.
Thus, [tex]\((f + g)(4)\)[/tex] can be written as:
[tex]\[ (f + g)(4) = f(4) + g(4) \][/tex]
Examining the given choices:
1. [tex]\(f(4) + g(4)\)[/tex] - This directly follows from our breakdown and is the correct expression.
2. [tex]\(f(x) + g(4)\)[/tex] - This is incorrect because it involves a function value dependent on [tex]\(x\)[/tex] and another fixed at 4.
3. [tex]\(f(4 + g(4))\)[/tex] - This is incorrect because it implies substituting 4 + g(4) into the function [tex]\(f\)[/tex], which is not what we need.
4. [tex]\(4(f(x) + g(x))\)[/tex] - This is incorrect as it scales the sum of functions by 4 and applies to all [tex]\(x\)[/tex], not just 4.
Therefore, the equivalent expression to [tex]\((f + g)(4)\)[/tex] is:
[tex]\[ f(4) + g(4) \][/tex]
