If [tex]\( f(x) = |x| + 9 \)[/tex] and [tex]\( g(x) = -6 \)[/tex], which describes the range of [tex]\( (f+g)(x) \)[/tex]?

A. [tex]\( (f+g)(x) \geq 3 \)[/tex] for all values of [tex]\( x \)[/tex]

B. [tex]\( (f+g)(x) \leq 3 \)[/tex] for all values of [tex]\( x \)[/tex]

C. [tex]\( (f+g)(x) \leq 6 \)[/tex] for all values of [tex]\( x \)[/tex]

D. [tex]\( (f+g)(x) \geq 6 \)[/tex] for all values of [tex]\( x \)[/tex]



Answer :

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].