To solve for [tex]\((x + g)(x)\)[/tex] given the functions [tex]\(f(x) = 2x + 3\)[/tex] and [tex]\(g(x) = x^2 - 8\)[/tex], we need to follow these steps:
1. Determine [tex]\(g(x)\)[/tex]:
Given [tex]\(g(x) = x^2 - 8\)[/tex].
2. Calculate [tex]\(x + g(x)\)[/tex]:
[tex]\[
x + g(x) = x + (x^2 - 8)
\][/tex]
[tex]\[
x + g(x) = x + x^2 - 8
\][/tex]
3. Substitute [tex]\(x + g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
First, we need to recognize that [tex]\((x + g)(x)\)[/tex] here means we are looking at [tex]\(x + g(x)\)[/tex] and not directly substituting into [tex]\(f(x)\)[/tex]. The expression given as [tex]\((x + g)(x)\)[/tex] simplifies to [tex]\(x + g(x)\)[/tex].
Expanding [tex]\(x + g(x)\)[/tex], we have already found it’s equal to:
[tex]\[
x + x^2 - 8
\][/tex]
Therefore, the function [tex]\((x + g)(x)\)[/tex] simplifies to [tex]\(x + x^2 - 8\)[/tex].
So, from the multiple choices provided:
- B. [tex]\(x^2 + 2x - 5\)[/tex]
- A. [tex]\(2x^3 - 18\)[/tex]
- C. [tex]\(2x^2 - 5\)[/tex]
- D. [tex]\(x^2 + 2x + 71\)[/tex]
The correct form does not match [tex]\(x + x^2 - 8\)[/tex] directly. Let's see which option closely matches by simple expansion and comparison steps:
Let's inspect each option.
For B:
[tex]\[
x^2 + 2x - 5
\][/tex]
This is not the same as [tex]\(x + x^2 - 8\)[/tex].
For A:
[tex]\[
2x^3 - 18
\][/tex]
This is clearly not the same.
For C:
[tex]\[
2x^2 - 5
\][/tex]
Again, it doesn’t resemble [tex]\(x + x^2 - 8\)[/tex].
For D:
[tex]\[
x^2 + 2x + 71
\][/tex]
Clearly, it doesn’t resemble [tex]\(x + x^2 - 8\)[/tex].
None match perfectly with [tex]\(x + x^2 - 8\)[/tex], but if you observe carefully, the right transformation function would most translate into,
Given options aren’t exactly fitting solving pattern, so no option clearly defines perfectly.
Thus without a given clear match, simplifies to:
[tex]\(
\boxed{x^2 + x - 8}
\)[/tex]
