Question

for what type of function will {g(f(x))}and {f(g(x))} both equal to x? explain.

Answer 1

When one is the inverse of the other.

Answer 2

y

Answer 3

Take the first case, g(f(x)). You take x and f maps it to f(x). You're wanting g now such that g will take that result and send it back to x (i.e. undo the operation). The function that will do that is the inverse of the original function.

Answer 4

This is the definition of an inverse function.

Answer 5

Example, \[f(x)=e^x , g(x) = \ln x\]then\[f(g(x))=e^{\ln x}=x\]by definition of logarithm, and\[g(f(x))=\ln e^x = x \ln e = x \]by log laws. The two functions,\[f(x)=e^x , g(x)=\ln x\]are inverse to each other. There is an infinity of such functions.