Problem
6. (1 point) If two functions are inverses of each other, what must happen when evaluating them symbolically? a. $\left(f \circ f^{-1}\right)(x)=0$ and $\left(f^{-1} \circ f\right)(x)=x$ b. $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=0$ c. $\left(f \circ f^{-1}\right)(x)=0$ and $\left(f^{-1} \circ f\right)(x)=0$ d. $\left(f \circ f^{-1}\right)(x)=x$ and $\left(f^{-1} \circ f\right)(x)=x$ e. none of these
Solution
Step 1 ：The question is asking about the properties of inverse functions. If two functions are inverses of each other, then the composition of the function and its inverse should return the original input. This means that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). This is because the inverse function 'undoes' the operation of the original function, returning the original input.
Step 2 ：Final Answer: The correct answer is \(\boxed{\text{(d) } \left(f \circ f^{-1}\right)(x)=x \text{ and } \left(f^{-1} \circ f\right)(x)=x}\).
