Checking an Inverse

A function and its inverse undo each other, so composing them gives back the input: $f\left({f}^{-1}\left(x\right)\right) = x$. That is a test for any claimed inverse: compose the two and see whether x comes back.

Take $p\left(x\right) = 4x-3$ and the claim $q\left(x\right) = \frac{x+3}{4}$. Then $p\left(q\left(x\right)\right) = \frac{4\cdot \left(x+3\right)}{4}-3$. The 4s cancel, leaving $x+3-3$, which is x. The claim passes.

A wrong guess fails in plain sight. Try $r\left(x\right) = \frac{x}{4}+3$: $p\left(r\left(x\right)\right) = x+12-3$, which is $x+9$, not x. The guess undid the steps in the wrong order: undoing times 4, then minus 3 means adding 3 first and dividing by 4 last.