To solve this problem, we will start by understanding the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], and then we will determine the function [tex]\( (f + g)(x) \)[/tex].
The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[ f(x) = |x| + 9 \][/tex]
The function [tex]\( g(x) \)[/tex] is defined as:
[tex]\[ g(x) = -6 \][/tex]
We need to find the combined function [tex]\( f + g \)[/tex]. Specifically, we need to find [tex]\( (f + g)(x) \)[/tex], which is given by:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
Substituting in the definitions of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f + g)(x) = (|x| + 9) + (-6) \][/tex]
[tex]\[ (f + g)(x) = |x| + 3 \][/tex]
Now, let's analyze the range of the function [tex]\( |x| + 3 \)[/tex].
The absolute value function [tex]\( |x| \)[/tex] always produces a non-negative output. This means:
[tex]\[ |x| \geq 0 \][/tex]
Adding 3 to both sides of this inequality gives:
[tex]\[ |x| + 3 \geq 3 \][/tex]
Thus, the function [tex]\( |x| + 3 \)[/tex] is always greater than or equal to 3 for all values of [tex]\( x \)[/tex].
Therefore, the range of the function [tex]\( (f + g)(x) = |x| + 3 \)[/tex] is all real numbers greater than or equal to 3.
So the correct description is:
[tex]\[ (f + g)(x) \geq 3 \ \text{for all values of} \ x \][/tex]
Hence, the correct answer is:
[tex]\[ (f+g)(x) \geq 3 \ \text{for all values of} \ x \][/tex]
