Finding an Inverse by Swapping and Solving

A function takes an input and produces an output. Its inverse reverses that: it takes the output and hands back the original input. Since every point (x, y) on f becomes the point (y, x) on its inverse, swapping x and y in the equation is exactly the move that flips input and output roles. Once you've swapped, the equation is written in terms of the new y, so solving for that y gives you the inverse formula, written f⁻¹(x).

The mechanism is just algebra on the swapped equation. Write y = f(x), trade the letters to get x = f(y), then isolate y using whatever steps undo the operations in order: undo addition and subtraction first, then multiplication, then powers or roots last, mirroring the reverse order of operations. For f(x) = (2x + 6)/5, swapping gives x = (2y + 6)/5; multiply by 5, subtract 6, divide by 2, and you get y = (5x - 6)/2. For g(x) = x³ - 4, swapping gives x = y³ - 4, so y = cube root of (x + 4); since cubing and cube-rooting both work for all real numbers, the domain is all reals.

This connects directly to transformations: an inverse is the graph of f reflected across the line y = x, which is why the swap works geometrically as well as algebraically. It also explains why not every function has a full inverse. f(x) = x² fails across all real numbers because two different inputs, like 3 and -3, give the same output 9 — swapping x and y produces x = y², and solving gives y = plus or minus √x, which is not a single function unless you restrict the domain.