Certainly! Let's carefully solve the problem step-by-step.
Given the function:
[tex]\[ g(x) = x - 6 \][/tex]
We need to determine its opposite, which is denoted as [tex]\( -g(x) \)[/tex].
1. Determine the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x - 6 \][/tex]
2. Calculate the opposite of [tex]\( g(x) \)[/tex]:
The opposite of [tex]\( g(x) \)[/tex] is defined as:
[tex]\[ -g(x) \][/tex]
3. Apply the negative sign to [tex]\( g(x) \)[/tex]:
If [tex]\( g(x) = x - 6 \)[/tex], then:
[tex]\[ -g(x) = -(x - 6) \][/tex]
4. Distribute the negative sign:
[tex]\[ -(x - 6) = -x + 6 \][/tex]
So, the opposite of [tex]\( g(x) = x - 6 \)[/tex] is:
[tex]\[ -g(x) = -x + 6 \][/tex]
5. Verify the result by evaluating [tex]\( g(x) \)[/tex] and [tex]\( -g(x) \)[/tex] at a specific value of [tex]\( x \)[/tex]:
Let's choose [tex]\( x = 0 \)[/tex]:
For [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = 0 - 6 = -6 \][/tex]
Now, for [tex]\( -g(x) \)[/tex]:
[tex]\[ -g(0) = -0 + 6 = 6 \][/tex]
Therefore, the opposite of the function [tex]\( g(x) = x - 6 \)[/tex], which is [tex]\( -g(x) \)[/tex], is indeed [tex]\( -x + 6 \)[/tex]. When evaluated at [tex]\( x = 0 \)[/tex], the results are:
[tex]\[ g(0) = -6 \][/tex]
[tex]\[ -g(0) = 6 \][/tex]
These computations confirm that our steps and final function [tex]\( -g(x) = -x + 6 \)[/tex] are correct.
