To solve the problem step by step, let's analyze the given functions and their operations.
1. Function Definition:
We start with the function [tex]\( g(x) = x - 6 \)[/tex].
2. Opposite Function:
The opposite of [tex]\( g(x) \)[/tex] is [tex]\( -g(x) \)[/tex]. By definition, this means multiplying the entire function [tex]\( g(x) \)[/tex] by [tex]\(-1\)[/tex].
3. Calculate [tex]\(-g(x)\)[/tex]:
[tex]\[
-g(x) = -(x - 6)
\][/tex]
Applying the negative sign inside the parentheses, we distribute:
[tex]\[
-g(x) = -x + 6
\][/tex]
4. Sum of [tex]\( g(x) \)[/tex] and [tex]\( -g(x) \)[/tex]:
Now we need to compute the sum of [tex]\( g(x) \)[/tex] and [tex]\( -g(x) \)[/tex]:
[tex]\[
g(x) + (-g(x)) = (x - 6) + (-x + 6)
\][/tex]
5. Combine Like Terms:
Let's add the terms:
[tex]\[
x - 6 - x + 6
\][/tex]
Combine [tex]\( x \)[/tex] and [tex]\(-x\)[/tex]:
[tex]\[
x - x = 0
\][/tex]
Combine [tex]\(-6\)[/tex] and [tex]\(6\)[/tex]:
[tex]\[
-6 + 6 = 0
\][/tex]
So the entire expression reduces to:
[tex]\[
0 + 0 = 0
\][/tex]
6. Final Result:
Therefore, the result of [tex]\( g(x) + (-g(x)) \)[/tex] is:
[tex]\[
0
\][/tex]
In conclusion, the sum [tex]\( g(x) + (-g(x)) \)[/tex] simplifies to [tex]\( 0 \)[/tex].
