If f(x) And its inverse function, what is your point of intersection

Question

If f(x) and its inverse function, f^-1(x), are both plotted on the same coordinate plane, what is their point of intersection?

Answer 1

The graph of y = f^-1(x) can be obtained from the graph of y = f(x) by interchanging coordinates. Geometrically, the graph of y = f^-1(x) is the reflection of the graph of y = f(x) in the line y = x. So, if the graph of y = f(x) intersects the line y = x, then that point of intersection will also be on the graph of

y = f^-1(x).

For example, if f(x) = x³, then f^-1(x) = ³√x. The graphs intersect at the point (1,1), a point on the line y = x.

On the other hand, if y = lnx, then f^-1(x) = e^x. The graph of f(x) = lnx doesn't intersect the line y = x, so f and f^-1 do not intersect.