To determine the range of the function [tex]\((f+g)(x)\)[/tex], we need to understand the individual behaviors of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] and how they combine.
#### Step 1: Analyze [tex]\(f(x)\)[/tex]
The function [tex]\(f(x)=|x|+9\)[/tex] is defined as:
- For [tex]\(x \geq 0\)[/tex], [tex]\(f(x) = x + 9\)[/tex]
- For [tex]\(x < 0\)[/tex], [tex]\(f(x) = -x + 9\)[/tex]
Since [tex]\(|x|\)[/tex] is always non-negative, the minimum value of [tex]\(|x| + 9\)[/tex] occurs when [tex]\(|x| = 0\)[/tex], which gives:
[tex]\[ f(0) = |0| + 9 = 9 \][/tex]
Therefore, [tex]\(f(x)\)[/tex] has a minimum value of 9 and increases as [tex]\(|x|\)[/tex] increases. So the range of [tex]\(f(x)\)[/tex] is:
[tex]\[ f(x) \geq 9 \][/tex]
#### Step 2: Analyze [tex]\(g(x)\)[/tex]
The function [tex]\(g(x) = -6\)[/tex] is a constant function, meaning it is always equal to -6 regardless of the value of [tex]\(x\)[/tex]:
[tex]\[ g(x) = -6 \text{ for all } x \][/tex]
#### Step 3: Combine [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]
We now sum [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] to get [tex]\((f+g)(x)\)[/tex]:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
[tex]\[ (f+g)(x) = (|x| + 9) + (-6) \][/tex]
[tex]\[ (f+g)(x) = |x| + 3 \][/tex]
#### Step 4: Determine the Range of [tex]\((f+g)(x)\)[/tex]
Since the minimum value of [tex]\(|x|\)[/tex] is 0, the minimum value of [tex]\(f(x) + g(x)\)[/tex] occurs when [tex]\(x = 0\)[/tex]:
[tex]\[ (f+g)(0) = |0| + 3 = 3 \][/tex]
As [tex]\(|x|\)[/tex] increases, [tex]\(|x| + 3\)[/tex] also increases without bound. Thus, the function [tex]\((f+g)(x)\)[/tex] ranges from:
[tex]\[ (f+g)(x) \geq 3 \][/tex]
Summarizing, the correct description of the range of [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ (f+g)(x) \geq 3 \text{ for all values of } x \][/tex]
The correct answer is:
[tex]\[
(f+g)(x) \geq 3 \text{ for all values of } x
\][/tex]
