Question

What is the reflective line of symmetry for an inverse function?

Answer 1

The line `y=x`

Explanation:

Given the graph of a function, the graph of its inverse relation is given by reflecting it in the line `y=x`. This may or may not be an inverse function.

For example, consider the function `y = x^2`, which has graph like this:

graph{x^2 [-10, 10, -5, 5]}

Reflecting this graph in the line `y=x` we get this graph:

graph{(x-y^2) = 0 [-10, 10, -5, 5]}

This relfected graph fails the vertical line test - so is not a function. For example, the line `x=1` cuts the horizontal parabola at two points: `(1, 1)` and `(1, -1)`:

graph{(x-y^2)(x-1+0.0001y)((x-1)^2+(y-1)^2-0.002)((x-1)^2+(y+1)^2-0.002) = 0 [-5.583, 4.417, -2.36, 2.64]}

By way of contrast, consider the function `y = x^3`:

graph{x^3 [-5.583, 4.417, -2.36, 2.64]}

Reflecting this in the line `y=x` we get the graph of the function `y = root(3)(x)`

graph{(y^3-x) = 0 [-5.583, 4.417, -2.36, 2.64]}