Let's solve the question step-by-step.
First, we are given the function [tex]\( g(x) = x - 6 \)[/tex].
### Step 1: Finding [tex]\(-g(x)\)[/tex]
The opposite of [tex]\( g(x) \)[/tex] is [tex]\(-g(x)\)[/tex]. To find [tex]\(-g(x)\)[/tex], we multiply [tex]\( g(x) \)[/tex] by [tex]\(-1\)[/tex]:
[tex]\[
-g(x) = -1 \cdot (x - 6)
\][/tex]
Distribute the [tex]\(-1\)[/tex] across the terms inside the parentheses:
[tex]\[
-g(x) = -x + 6
\][/tex]
So, we have:
[tex]\[
-g(x) = 6 - x
\][/tex]
### Step 2: Adding [tex]\( g(x) \)[/tex] and [tex]\(-g(x)\)[/tex]
Next, we need to find the sum of [tex]\( g(x) \)[/tex] and [tex]\(-g(x) \)[/tex]:
[tex]\[
g(x) + (-g(x)) = (x - 6) + (6 - x)
\][/tex]
Combine the terms:
[tex]\[
(x - 6 + 6 - x)
\][/tex]
Notice that [tex]\( x \)[/tex] and [tex]\(-x\)[/tex] cancel each other out:
[tex]\[
x - x + 6 - 6 = 0
\][/tex]
So, the sum is:
[tex]\[
g(x) + (-g(x)) = 0
\][/tex]
### Final Answer
[tex]\[
-g(x) = 6 - x
\][/tex]
[tex]\[
g(x) + (-g(x)) = 0
\][/tex]
