Let's solve the given problem step by step.
First, we have the function [tex]\( g(x) = x - 6 \)[/tex].
Step 1: Opposite of [tex]\( g(x) \)[/tex]
To find the opposite of [tex]\( g(x) \)[/tex], we simply negate the function:
[tex]\[ -g(x) = -(x - 6) \][/tex]
Distributing the negative sign, we get:
[tex]\[ -g(x) = -x + 6 \][/tex]
Therefore:
[tex]\[ -g(x) = -x + 6 \][/tex]
Step 2: Sum of [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]\[ g(x) + (-g(x)) \][/tex]
Substituting the given expressions, we have:
[tex]\[ g(x) = x - 6 \][/tex]
[tex]\[ -g(x) = -x + 6 \][/tex]
Now, adding these two together:
[tex]\[ g(x) + (-g(x)) = (x - 6) + (-x + 6) \][/tex]
Combining like terms:
[tex]\[ g(x) + (-g(x)) = (x + (-x)) + (-6 + 6) \][/tex]
[tex]\[ g(x) + (-g(x)) = 0 + 0 \][/tex]
[tex]\[ g(x) + (-g(x)) = 0 \][/tex]
So, to summarize:
1. [tex]\( -g(x) = -x + 6 \)[/tex]
2. [tex]\( g(x) + (-g(x)) = 0 \)[/tex]
Thus, the final solution is:
[tex]\[ -g(x) = -x + 6 \][/tex]
[tex]\[ g(x) + (-g(x)) = 0 \][/tex]
