Final answer:
By composing the functions f(x) and g(x), and simplifying the result, we find that f(g(x)) simplifies to x, confirming that g(x) is the inverse of f(x). Therefore, the statement is true.
Step-by-step explanation:
The question asks whether the functions f(x) = \frac{x}{2} + 3 and g(x) = 2x - 6 are inverses of each other. To determine if two functions are inverses, we can compose them (i.e., substitute one function into the other) and see if the result is the identity function, which returns the original input (x). Let's check if f(g(x)) equals x.
First, apply g(x) to x, which gives g(x) = 2x - 6. Now substitute g(x) into f(x), so f(g(x)) = f(2x - 6) = \frac{1}{2}(2x - 6) + 3.
Simplify f(g(x)):
- f(g(x)) = \frac{1}{2}(2x - 6) + 3
- = x - 3 + 3
- = x
Since f(g(x)) simplifies to x, this means that g(x) is indeed the inverse of f(x). Hence, the statement is True.