To solve the problem, let's start by defining the functions and determining their combined effect.
1. Given functions:
- [tex]\( f(x) = |x| + 9 \)[/tex]
- [tex]\( g(x) = -6 \)[/tex]
2. Sum of the functions:
[tex]\[
(f + g)(x) = f(x) + g(x)
\][/tex]
Substituting the given functions:
[tex]\[
(f + g)(x) = (|x| + 9) + (-6)
\][/tex]
Simplifying:
[tex]\[
(f + g)(x) = |x| + 9 - 6 = |x| + 3
\][/tex]
3. Analyzing the expression [tex]\( |x| + 3 \)[/tex]:
- [tex]\( |x| \)[/tex] represents the absolute value of [tex]\( x \)[/tex], which is always non-negative. That means [tex]\( |x| \geq 0 \)[/tex] for all [tex]\( x \)[/tex].
- Therefore, the minimum value of [tex]\( |x| \)[/tex] is [tex]\( 0 \)[/tex].
- Adding [tex]\( 3 \)[/tex] to [tex]\( |x| \)[/tex], we get the minimum value of [tex]\( |x| + 3 = 0 + 3 = 3 \)[/tex].
4. Range of [tex]\( (f + g)(x) \)[/tex]:
- Because [tex]\( |x| \geq 0 \)[/tex], the expression [tex]\( |x| + 3 \)[/tex] will always be greater than or equal to [tex]\( 3 \)[/tex].
- Hence, [tex]\( (f + g)(x) \geq 3 \)[/tex] for all values of [tex]\( x \)[/tex].
5. Conclusion:
The correct option that describes the range of [tex]\( (f + g)(x) \)[/tex] is:
[tex]\[
(f+g)(x) \geq 3 \text{ for all values of } x
\][/tex]
So, the answer is that the range of [tex]\( (f+g)(x) \)[/tex] is given by:
[tex]\[
\boxed{(f+g)(x) \geq 3 \text{ for all values of } x}
\][/tex]
