An inverse reverses input and output. To find it, write y=f(x), rearrange for x, and then express the reversed rule using x as its input. An inverse is a function only if each output of the original comes from one permitted input. Example For y=3x-5, x= y+53, so f-1(x)= x+53. Check f-1(f(4))=f-1(7)=4. The inverse undoes the multiplication and subtraction in reverse order. For g(x)=x2 restricted to x 0, the inverse is g-1(x)= x, with x 0. On all real inputs, the square rule sends both 2 and −2 to 4, so reversing it would not give a unique output. The notation f-1 means the inverse rule, not 1f. The inverse swaps the original domain and range.

An inverse reverses input and output. To find it, write \(y=f(x)\), rearrange for x, and then express the reversed rule using x as its input. An inverse is a function only if each output of the original comes from one permitted input.

For \(y=3x-5\), \(x=\dfrac{y+5}{3}\), so \(f^{-1}(x)=\dfrac{x+5}{3}\). Check \(f^{-1}(f(4))=f^{-1}(7)=4\). The inverse undoes the multiplication and subtraction in reverse order.

For \(g(x)=x^2\) restricted to \(x\ge0\), the inverse is \(g^{-1}(x)=\sqrt x\), with \(x\ge0\). On all real inputs, the square rule sends both 2 and −2 to 4, so reversing it would not give a unique output.

The notation \(f^{-1}\) means the inverse rule, not \(\dfrac1f\). The inverse swaps the original domain and range.

Functions

Cambridge IGCSE Mathematics 0580. Initial catalogue covers the assessed demands of February/March 2026 Paper 22; further Core and Extended mappings require paper-level scope review.

Undoing a function

Undoing a function