Inverse functions and their graphs

If y = f(x), the inverse f⁻¹ satisfies f⁻¹(y) = x. Applying one after the other gets you back where you started: f⁻¹(f(x)) = x. The notation f⁻¹ means the inverse function, not 1/f(x).

Every point (a, b) on the graph of f becomes (b, a) on the graph of f⁻¹. Swapping coordinates is the same as reflecting in the line y = x. So to sketch an inverse, reflect the original: (0, 1) on y = 2ˣ becomes (1, 0) on y = log₂ x, and the horizontal asymptote y = 0 becomes the vertical asymptote x = 0.

1. Write y = f(x). For example, y = (2x + 3)/5.
2. Swap x and y: x = (2y + 3)/5.
3. Solve for y: 5x = 2y + 3, so y = (5x − 3)/2.
4. That is f⁻¹(x). Check: f(f⁻¹(x)) = (2·(5x − 3)/2 + 3)/5 = x.

A function has an inverse only if it is one-to-one: no two inputs give the same output. On a graph, that means no horizontal line crosses it more than once. y = x³ passes; y = x² fails, because 2 and −2 both give 4, so “undo squaring” has two answers.

The fix is to restrict the domain. On x ≥ 0, x² is one-to-one, and its inverse is √x. The same trick defines the inverse trig functions: sine restricted to −π/2 ≤ x ≤ π/2 has inverse arcsin.

• The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.
• A graph and its inverse cross, if at all, on the line y = x or in pairs symmetric about it.
• Slopes are reciprocal: if f has slope m at (a, b), then f⁻¹ has slope 1/m at (b, a). A horizontal tangent becomes a vertical one, which is why √x is vertical at the origin.