Final answer:
Composite functions involve substituting one function into another. For f(g(x)), g(f(x)), f(f(x)), and g(g(x)), we found expressions and confirmed that the domain for each is all real numbers because there are no restrictions from the operations used in the compositions.
Step-by-step explanation:
To solve for the composite functions f(g(x)) and g(f(x)), along with f(f(x)) and g(g(x)), we need to substitute one function into the other. The domain of a composite function is the set of all input values, or x-values, that can be used for which the composite function is defined.
Composite Functions:
- f(g(x)): This means plugging g(x) into f(x). Starting with g(x) = x^2, when we substitute x^2 into f(x), we get f(g(x)) = f(x^2) = 2(x^2) + 7. The domain is all real numbers since there's no restriction on squaring a real number.
- g(f(x)): This means plugging f(x) into g(x). From f(x) = 2x + 7, when we substitute 2x + 7 into g(x), we get g(f(x)) = g(2x + 7) = (2x + 7)^2. The domain is also all real numbers because any real number can be plugged into f(x) and then squared.
- f(f(x)): This means plugging f(x) into itself. So f(f(x)) = f(2x + 7) = 2(2x + 7) + 7 = 4x + 21. The domain for this is all real numbers.
- g(g(x)): Similarly, this is plugging g(x) into itself. So g(g(x)) = g(x^2) = (x^2)^2 = x^4. The domain for g(g(x)) is all real numbers as well.