Question

The graph of f(x) is shown below.

Answer 1

The intersection would be at the point (2, 2).

This is because, graphically, the plots of f(x) and its inverse are reflections of one another across the line y = x, and (2, 2) lies on this line.

Put another way, we have f(2) = 2 = f⁻¹(2), so both f(x) and f⁻¹(x) intersect when x = 2.

Put yet another (longer) way, we can find the equation for f(x): it's a line that passes through (0, 6) and (3, 0), so it has slope -6/3 = -2. Then using the point-slope formula,

y - 6 = -2 (x - 0)   ⇒   y = f(x) = -2x + 6

By definition of function inverse, we have

f(f⁻¹(x)) = x

so that with the given definition of f(x), we get

f(f⁻¹(x)) = -2 f⁻¹(x) + 6 = x

-2 f⁻¹(x) = x - 6

f⁻¹(x) = -x/2 + 3

Then we solve for x such that f(x) = f⁻¹(x). We would find x = 2 as before.