To determine the range of [tex]\((f + g)(x)\)[/tex], we first need to understand the behavior of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] individually, and then how they combine.
### Step-by-Step Solution:
1. Function Definitions:
- [tex]\( f(x) = |x| + 9 \)[/tex]
- [tex]\( g(x) = -6 \)[/tex]
2. Combining the Functions:
- The combined function [tex]\((f + g)(x)\)[/tex] is given by:
[tex]\[
(f + g)(x) = f(x) + g(x)
\][/tex]
3. Substitute the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- Since [tex]\( f(x) = |x| + 9 \)[/tex] and [tex]\( g(x) = -6 \)[/tex], we substitute these into the combined function:
[tex]\[
(f + g)(x) = (|x| + 9) + (-6)
\][/tex]
4. Simplify the Expression:
- Simplifying the combined expression:
[tex]\[
(f + g)(x) = |x| + 9 - 6
\][/tex]
- This reduces to:
[tex]\[
(f + g)(x) = |x| + 3
\][/tex]
5. Analyze the Expression:
- We now need to determine the range of the function [tex]\( |x| + 3 \)[/tex]. Recall that [tex]\( |x| \)[/tex] (the absolute value of [tex]\( x \)[/tex]) is always non-negative, which means [tex]\( |x| \geq 0 \)[/tex] for all [tex]\( x \)[/tex].
- Therefore, the minimum value of [tex]\( |x| \)[/tex] is 0 (when [tex]\( x = 0 \)[/tex]):
[tex]\[
|x| + 3 \geq 0 + 3
\][/tex]
- So, [tex]\( |x| + 3 \geq 3 \)[/tex] for all values of [tex]\( x \)[/tex].
6. Range of [tex]\((f + g)(x)\)[/tex]:
- From this analysis, we can conclude that:
[tex]\[
(f+g)(x) \geq 3 \text{ for all values of } x
\][/tex]
### Conclusion:
The range of [tex]\((f + g)(x)\)[/tex] is described by the statement:
[tex]\[
(f + g)(x) \geq 3 \text{ for all values of } x
\][/tex]
Therefore, the correct answer is: [tex]\((f + g)(x) \geq 3\)[/tex] for all values of [tex]\( x \)[/tex].
