To find [tex]\((f + g)(x)\)[/tex] given the functions [tex]\(f(x) = 5 - x\)[/tex] and [tex]\(g(x) = x^2 + 3x - 2\)[/tex], we need to add the two functions together.
1. Write down the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = 5 - x
\][/tex]
[tex]\[
g(x) = x^2 + 3x - 2
\][/tex]
2. To find [tex]\((f + g)(x)\)[/tex], add the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f + g)(x) = f(x) + g(x)
\][/tex]
[tex]\[
(f + g)(x) = (5 - x) + (x^2 + 3x - 2)
\][/tex]
3. Combine like terms:
[tex]\[
(f + g)(x) = 5 - x + x^2 + 3x - 2
\][/tex]
4. Arrange the terms in descending order of [tex]\(x\)[/tex]:
[tex]\[
(f + g)(x) = x^2 + 3x - x + 5 - 2
\][/tex]
5. Simplify the expression by combining like terms:
[tex]\[
(f + g)(x) = x^2 + 2x + 3
\][/tex]
So, the function [tex]\((f + g)(x)\)[/tex] is:
[tex]\[
(f + g)(x) = x^2 + 2x + 3
\][/tex]
To exemplify the result, let's use [tex]\(x = 1\)[/tex].
6. Substitute [tex]\(x = 1\)[/tex] into [tex]\((f + g)(x)\)[/tex]:
[tex]\[
(f + g)(1) = 1^2 + 2(1) + 3
\][/tex]
[tex]\[
(f + g)(1) = 1 + 2 + 3
\][/tex]
[tex]\[
(f + g)(1) = 6
\][/tex]
Hence, when [tex]\(x = 1\)[/tex], the value of [tex]\((f + g)(x)\)[/tex] is 6.
