To find [tex]\((f+g)(x)\)[/tex] given the functions [tex]\(f(x) = 4x + 2\)[/tex] and [tex]\(g(x) = x^2 - 6\)[/tex], we need to combine these functions by adding them together. Here’s a detailed, step-by-step solution:
1. Define the functions:
- [tex]\(f(x) = 4x + 2\)[/tex]
- [tex]\(g(x) = x^2 - 6\)[/tex]
2. Combine the two functions:
- [tex]\((f+g)(x) = f(x) + g(x)\)[/tex]
3. Substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f+g)(x) = (4x + 2) + (x^2 - 6)
\][/tex]
4. Combine like terms:
[tex]\[
(f+g)(x) = x^2 + 4x + 2 - 6
\][/tex]
5. Simplify the expression:
[tex]\[
(f+g)(x) = x^2 + 4x - 4
\][/tex]
Now let's verify our result with an example value of [tex]\(x\)[/tex]. Let's choose [tex]\(x = 1\)[/tex]:
6. Evaluate [tex]\(f(1)\)[/tex]:
[tex]\[
f(1) = 4(1) + 2 = 4 + 2 = 6
\][/tex]
7. Evaluate [tex]\(g(1)\)[/tex]:
[tex]\[
g(1) = (1)^2 - 6 = 1 - 6 = -5
\][/tex]
8. Evaluate [tex]\((f+g)(1)\)[/tex]:
[tex]\[
(f+g)(1) = f(1) + g(1) = 6 + (-5) = 1
\][/tex]
Therefore, the combined function [tex]\( (f + g)(x) \)[/tex] is:
[tex]\[
(f+g)(x) = x^2 + 4x - 4
\][/tex]
When we evaluate it for [tex]\( x = 1 \)[/tex], we get:
[tex]\((f+g)(1) = 1\)[/tex].
This confirms that our function [tex]\((f+g)(x) = x^2 + 4x - 4\)[/tex] is correct.
