Question

Which of the following are true about inverse functions? a) The domain and the range are interchanged from the original function to the inverse function. b) If the original is a function, then the inverse is always a function. c) (f^(-1) ∘ f)(x) = (f ∘ f^(-1))(x) d) The original and the inverse are reflections over the line y = x.

Answer 1

All of the statements (A, B, C, and D) are true about inverse functions.

A. True. The domain and range of the inverse function are indeed swapped compared to the original function.

B. True. If a function has an inverse, then the inverse is also a function. However, not all functions have inverses.

C. True. This statement expresses the property of inverse functions. The composition of a function with its inverse (in either order) results in the identity function, which is represented as
`$\left(f^(-1) \circ f\right)(x) = x = \left(f \circ f^(-1)\right)(x)$`.

D. True. The graph of the original function and its inverse are reflections of each other across the line
`$y=x$`.