To solve this problem, let's delve into the given expression step by step.
1. Understand the given function:
We are provided with the function:
[tex]\[
g(x) = x - 6
\][/tex]
2. Determine the opposite of [tex]\(g(x)\)[/tex]:
The problem asks us to find the opposite of [tex]\(g(x)\)[/tex], which is denoted as [tex]\(-g(x)\)[/tex]. According to the rules of algebra, when we take the negative of a function [tex]\(g(x)\)[/tex], we essentially multiply the entire function by [tex]\(-1\)[/tex].
3. Apply the negative sign to the given function [tex]\(g(x)\)[/tex]:
Starting with the function [tex]\(g(x)\)[/tex]:
[tex]\[
g(x) = x - 6
\][/tex]
Now, multiply this expression by [tex]\(-1\)[/tex] to find [tex]\(-g(x)\)[/tex]:
[tex]\[
-g(x) = -1 \cdot (x - 6)
\][/tex]
4. Distribute the negative sign across the terms inside the parentheses:
[tex]\[
-g(x) = -x + 6
\][/tex]
So, the opposite of the function [tex]\(g(x)\)[/tex] is:
[tex]\[
-g(x) = -x + 6
\][/tex]
5. Verify the result at [tex]\(x = 1\)[/tex]:
To ensure the correctness of our derivation, let's verify by substituting [tex]\(x = 1\)[/tex]:
[tex]\[
g(1) = 1 - 6 = -5
\][/tex]
[tex]\[
-g(1) = -(-5) = 5
\][/tex]
Thus, the final expression for [tex]\(-g(x)\)[/tex] is:
[tex]\[
-g(x) = -x + 6
\][/tex]
