Final answer:
To solve the function composition, we plug g(x) into f(x), yielding the simplified expression -5x + 13/3. The inverse function g^(-1)(21) is found to be 5. The domain of f(x) is all real numbers, as there are no operations that restrict the domain.
Step-by-step explanation:
The student has asked for assistance with a function composition problem, an inverse function problem, and determining the domain of a function in the subject of mathematics. The questions relate to the functions g(x) = 5x - 4 and f(x) = 1/3 - x.
Function Composition
f(g(x)) means plugging g(x) into f(x). Starting with f(g(x)) = f(5x - 4), we substitute 5x - 4 in place of x in f(x), yielding f(5x - 4) = 1/3 - (5x - 4). Simplifying, f(g(x)) = 1/3 - 5x + 4 or f(g(x)) = -5x + 4 + 1/3 which simplifies to -5x + 13/3.
Inverse Function
To find g^{-1}(21), we solve the equation g(x) = 21 for x. This gives us 5x - 4 = 21. Adding 4 to both sides, 5x = 25, and dividing by 5 yields x = 5. Therefore, g^{-1}(21) = 5.
Domain of f(x)
The domain of f(x) is all real numbers, since there are no restrictions such as square roots or variables in the denominator that would exclude any real numbers.