Final answer:
The compositions (f∘g)(x) and (g∘f)(x) both result in the identity function x with domains being all real numbers.
Step-by-step explanation:
The student has asked how to find the composition of two functions, f(x)=x+12 and g(x)=x-12, and to determine the domains of each composition. The composition of two functions is the application of one function to the results of another. To find (f∘g)(x), we substitute g(x) into f(x):
(f∘g)(x) = f(g(x)) = f(x-12) = (x-12) + 12 = x.
The domain of (f∘g)(x) is all real numbers, as there are no restrictions on x in either function that would affect the composition.
Next, to find (g∘f)(x), we substitute f(x) into g(x):
(g∘f)(x) = g(f(x)) = g(x+12) = (x+12) - 12 = x.
The domain of (g∘f)(x) is also all real numbers for the same reason.
Interestingly, both compositions result in the identity function x, and both have the entire set of real numbers as their domain.