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?

On a coordinate plane, a curve opens down and to the right in quadrants 3 and 4 and then changes direction and curves up and to the left in quadrants 1 and 4. The curve crosses the y-axis at (0, negative 2), changes direction at (1, negative 1, and crosses the x-axis at (2, 0).

Answer 1

The graphs of a function and its inverse are symmetric about the line y = x, so any intersection point must satisfy f(x) = x.

The three given points (0, −2), (1, −1), (2, 0) lie on the cubic
f(x) = (x − 1)^3 − 1
(which has an inflection/change-of-direction at (1, −1) and passes through the other two points). Set f(x) = x:
(x − 1)^3 − 1 = x
=> (x − 1)^3 = x + 1.
Let t = x − 1, then t^3 − t − 2 = 0. By Cardano the real root is
t = ³√(1 + √(26/27)) + ³√(1 − √(26/27)),
so
x = 1 + ³√(1 + √(26/27)) + ³√(1 − √(26/27)) ≈ 2.52138.

Thus the unique intersection point is approximately (2.5214, 2.5214).