Sure, let's find the combined function [tex]\((f+g)(x)\)[/tex] given the functions [tex]\(f(x) = 3 + 8x\)[/tex] and [tex]\(g(x) = 5x - 2\)[/tex].
We need to add these two functions together to find [tex]\((f+g)(x)\)[/tex].
Step by step:
1. Start by writing down the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = 3 + 8x
\][/tex]
[tex]\[
g(x) = 5x - 2
\][/tex]
2. Now add [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f+g)(x) = f(x) + g(x)
\][/tex]
Substitute the expressions of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] into the equation:
[tex]\[
(f+g)(x) = (3 + 8x) + (5x - 2)
\][/tex]
3. Combine the like terms:
[tex]\[
(f+g)(x) = 3 + 8x + 5x - 2
\][/tex]
4. Combine the constants and the terms involving [tex]\(x\)[/tex]:
[tex]\[
(f+g)(x) = (3 - 2) + (8x + 5x)
\][/tex]
Simplify the constants and the coefficients of [tex]\(x\)[/tex]:
[tex]\[
(f+g)(x) = 1 + 13x
\][/tex]
Thus, the combined function:
[tex]\((f+g)(x) = 1 + 13x\)[/tex]
