Let's solve the problem step-by-step.
We are given two functions:
[tex]\[ f(x) = |x| + 9 \][/tex]
[tex]\[ g(x) = -6 \][/tex]
We need to determine the range of the sum [tex]\( (f+g)(x) \)[/tex].
First, calculate [tex]\( (f+g)(x) \)[/tex]:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
Substitute the given functions:
[tex]\[ f(x) = |x| + 9 \][/tex]
[tex]\[ g(x) = -6 \][/tex]
[tex]\[ (f+g)(x) = (|x| + 9) + (-6) = |x| + 3 \][/tex]
Now, we need to determine the range of the function [tex]\( |x| + 3 \)[/tex].
Recall that [tex]\( |x| \geq 0 \)[/tex] for all real numbers [tex]\( x \)[/tex]. Therefore,
[tex]\[ |x| \geq 0 \][/tex]
Adding 3 to both sides, we get:
[tex]\[ |x| + 3 \geq 3 \][/tex]
This means that the minimum value of [tex]\( |x| + 3 \)[/tex] is 3. Since [tex]\( |x| \)[/tex] can take any value from 0 to [tex]\(\infty\)[/tex], [tex]\( |x| + 3 \)[/tex] can take any value from 3 upwards to [tex]\(\infty\)[/tex].
Hence, the range of [tex]\( (f+g)(x) \)[/tex] is all real numbers greater than or equal to 3.
Therefore, the correct description of the range is:
[tex]\[ (f+g)(x) \geq 3 \text{ for all values of } x \][/tex]
Thus, the answer is:
[tex]\[ (f+g)(x) \geq 3 \text{ for all values of } x \][/tex]
The closest approximation or answer in the given multiple-choice options is:
[tex]\[ (f+g)(x) \geq 6 \text{ for all values of } x \][/tex]
