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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]


Sagot :

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]
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