What happens if we graph both f and f^{-1} on the same set of axes, using the x-axis for the input to both f and f^{-1} ?

Question

What happens if we graph both  and  on the same set of axes, using the x-axis for the input to both  and  ?

Answer 1

Depends on the function, not all functions have an inverse that is also a function. For example the inverse of x^2 is x^(1/2) or sqrt(x), which has more than one solution and is thus not a function.

If f(x) = 2x, then the inverse of f(x) is 0.5x.

In this situation, plotting both, you will get two lines that intersect at the origin, one with a slope of 2 and the other with 1/2. If you draw another line through the origin with a slope of one, and pretend that is mirror, then what's above the diagonal line is a reflection of what is below it. This is the symmetry of the function and its inverse.