To find g(f(x)), we first need to determine what f(x) is. We are given that:
f(x) = 3x + 6
Next, we will substitute f(x) into g(x). The function g(x) is defined as:
g(x) = x + 2
Now, to find g(f(x)), we replace x in g(x) with f(x):
g(f(x)) = g(3x + 6)
Now substituting into g(x):
g(f(x)) = (3x + 6) + 2
Simplifying this gives us:
g(f(x)) = 3x + 8
Now, let’s determine the domain of g(f(x)). The function f(x) is a linear function, and it is defined for all real numbers x. Similarly, since g(x) is also a linear function, it too is defined for all real numbers. Therefore, the output of g(f(x)) is also defined for all real numbers.
Thus, the domain of g(f(x)) is all real numbers.