An inverse function reverses an input-output pairing. If f sends 4 to 9, its inverse sends 9 to 4. The domain of the inverse is the range of the original function, and the range of the inverse is the original domain. Inverse notation means reverse the mapping, not take the reciprocal of its value.
Look for repeated outputs before solving
An injective function sends distinct inputs to distinct outputs. This is the horizontal-line test on a graph. A surjective function reaches every value of the stated codomain. When both conditions hold, the function is bijective and has an inverse from that codomain. An injective function can be inverted on its actual range even when its stated codomain is larger.
Undo multiplication and subtraction in reverse order
Let f multiply its input by 3 and subtract 5. To recover the input from an output y, add 5 first, then divide by 3. The inverse takes 7 to 4, and f takes 4 back to 7. These steps work on all real numbers. Substitution confirms both directions rather than just one numerical pair.
Restrict a square before trying to invert it
Squaring sends 2 and negative 2 to the same output, 4. An inverse on all real inputs would have to send 4 to two different numbers. If the original domain is restricted to nonnegative inputs, each output is reached once and the inverse is the nonnegative square root. Restricting to nonpositive inputs would give the negative of that square root instead.
