To determine the range of the function [tex]\((f+g)(x)\)[/tex], we start by examining the given functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Given:
[tex]\[ f(x) = |x| + 9 \][/tex]
[tex]\[ g(x) = -6 \][/tex]
We need to find the combined function [tex]\( (f+g)(x) \)[/tex].
First, compute [tex]\( (f+g)(x) \)[/tex]:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
[tex]\[ (f+g)(x) = (|x| + 9) + (-6) \][/tex]
[tex]\[ (f+g)(x) = |x| + 9 - 6 \][/tex]
[tex]\[ (f+g)(x) = |x| + 3 \][/tex]
Next, we need to determine the range of [tex]\(|x| + 3\)[/tex].
Consider the nature of the absolute value function [tex]\(|x|\)[/tex]:
- The smallest value [tex]\(|x|\)[/tex] can take is [tex]\(0\)[/tex], when [tex]\(x = 0\)[/tex].
- For any real number [tex]\(x\)[/tex], [tex]\(|x|\)[/tex] is always non-negative, meaning [tex]\(|x| \geq 0\)[/tex].
Therefore:
[tex]\[ |x| \geq 0 \][/tex]
[tex]\[ |x| + 3 \geq 0 + 3 \][/tex]
[tex]\[ |x| + 3 \geq 3 \][/tex]
This implies that the minimum value of [tex]\(|x| + 3\)[/tex] is 3, and as [tex]\(x\)[/tex] varies over all real numbers, [tex]\(|x| + 3\)[/tex] will likewise vary over all values greater than or equal to 3. Hence, the range of [tex]\((f+g)(x)\)[/tex] is all values greater than or equal to 3.
Therefore, the correct answer is:
[tex]\[ (f+g)(x) \geq 3 \text{ for all values of } x \][/tex]
