To determine the range of [tex]\((f+g)(x)\)[/tex] given the functions [tex]\( f(x) = |x| + 9 \)[/tex] and [tex]\( g(x) = -6 \)[/tex], let's find [tex]\((f+g)(x)\)[/tex] step-by-step.
1. Function Definitions:
[tex]\( f(x) = |x| + 9 \)[/tex]
[tex]\( g(x) = -6 \)[/tex]
2. Sum of Functions:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
3. Substitute the Definitions:
[tex]\[ (f+g)(x) = (|x| + 9) + (-6) \][/tex]
4. Simplify [tex]\( (f+g)(x) \)[/tex]:
[tex]\[ (f+g)(x) = |x| + 9 - 6 \][/tex]
[tex]\[ (f+g)(x) = |x| + 3 \][/tex]
5. Determine the Range of [tex]\((f+g)(x)\)[/tex]:
The absolute value function [tex]\( |x| \)[/tex] is always non-negative for all [tex]\( x \)[/tex]. Thus, [tex]\( |x| \geq 0 \)[/tex].
Adding 3 to [tex]\( |x| \)[/tex], we have [tex]\( |x| + 3 \geq 3 \)[/tex].
Therefore, the expression [tex]\(|x| + 3\)[/tex] takes values starting from 3 and increasing without bound as [tex]\( |x| \)[/tex] increases. This implies that:
[tex]\[
(f+g)(x) \geq 3 \text{ for all values of } x.
\][/tex]
So, the range of [tex]\((f+g)(x)\)[/tex] is best described by:
[tex]\[ \boxed{(f+g)(x) \geq 3 \text{ for all values of } x} \][/tex]
