Reflecting Over the Line y = x

An inverse function undoes what the original did, and that undoing shows up visually as a reflection. If (p, q) is a point on f, it means f sent p to q. The inverse reverses that: it sends q back to p. So the point (q, p) sits on the graph of f^−1. Swapping the coordinates of every point is exactly what reflecting across the line y = x does, since that line is where x and y coordinates are equal and swapping them is a mirror flip over it. That is the whole mechanism: build f⁻¹ by swapping x and y everywhere, and graphically that swap is a reflection.

Try f(x) = 2^x. The point (0, 1) lies on it, since 2⁰ = 1. The point (3, 8) lies on it too, since 2³ = 8. Reflect those across y = x and you get (1, 0) and (8, 3), which lie on the inverse, the logarithm log₂(x). Trace the whole curve this way and the rising exponential curve becomes the slower-rising log curve, bent the other way across the diagonal.

This reflection forces domain and range to trade places. Whatever x-values f could accept become the y-values f⁻¹ produces, and whatever y-values f could produce become the x-values f⁻¹ accepts. That trade is why 2^x, with domain all reals and range y > 0, has an inverse log₂(x) with domain x > 0 and range all reals. Keeping track of that swap is what makes the next topics, finding inverses algebraically and matching exponential models to log models, make sense.