Let's tackle the given problem step-by-step.
### Part a: (f + g)(x)
To find [tex]\((f + g)(x)\)[/tex], we need to add the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Given:
[tex]\[ f(x) = x^2 - 5 \][/tex]
[tex]\[ g(x) = 5x + 4 \][/tex]
First, sum them up:
[tex]\[
(f + g)(x) = f(x) + g(x) = (x^2 - 5) + (5x + 4)
\][/tex]
Combine like terms:
[tex]\[
(f + g)(x) = x^2 + 5x - 1
\][/tex]
### Part b: (f + g)(3)
To find [tex]\((f + g)(3)\)[/tex], substitute [tex]\(x = 3\)[/tex] into the expression found in part a.
Given:
[tex]\[
(f + g)(x) = x^2 + 5x - 1
\][/tex]
Substitute [tex]\(x = 3\)[/tex]:
[tex]\[
(f + g)(3) = 3^2 + 5(3) - 1
\][/tex]
[tex]\[
(f + g)(3) = 9 + 15 - 1
\][/tex]
[tex]\[
(f + g)(3) = 23
\][/tex]
So,
[tex]\((f + g)(3) = 23\)[/tex]
### Part c: (f - g)(x)
To find [tex]\((f - g)(x)\)[/tex], we need to subtract the function [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex].
Given:
[tex]\[ f(x) = x^2 - 5 \][/tex]
[tex]\[ g(x) = 5x + 4 \][/tex]
Subtract them:
[tex]\[
(f - g)(x) = f(x) - g(x) = (x^2 - 5) - (5x + 4)
\][/tex]
Distribute the negative sign:
[tex]\[
(f - g)(x) = x^2 - 5 - 5x - 4
\][/tex]
Combine like terms:
[tex]\[
(f - g)(x) = x^2 - 5x - 9
\][/tex]
### Part d: (f - g)(5)
To find [tex]\((f - g)(5)\)[/tex], substitute [tex]\(x = 5\)[/tex] into the expression found in part c.
Given:
[tex]\[
(f - g)(x) = x^2 - 5x - 9
\][/tex]
Substitute [tex]\(x = 5\)[/tex]:
[tex]\[
(f - g)(5) = 5^2 - 5(5) - 9
\][/tex]
[tex]\[
(f - g)(5) = 25 - 25 - 9
\][/tex]
[tex]\[
(f - g)(5) = -9
\][/tex]
So,
[tex]\((f - g)(5) = -9\)[/tex]
### Summary
- [tex]\((f + g)(x) = x^2 + 5x - 1\)[/tex]
- [tex]\((f + g)(3) = 23\)[/tex]
- [tex]\((f - g)(x) = x^2 - 5x - 9\)[/tex]
- [tex]\((f - g)(5) = -9\)[/tex]
These are the solutions for the given parts a, b, c, and d.
