Inverse Relations and Domain Restrictions

Inverse Relations and Domain Restrictions

An inverse reverses a function’s input-output process. Swap $x$ and $y$, then solve for $y$. A relation has an inverse function only when it is one-to-one on the chosen domain; otherwise restrict the domain. Graphs of inverses reflect in the line $y=x$, so domain and range swap.

Worked reasoning

For $y=3x-5$, swap: $x=3y-5$. Then $x+5=3y$, so $f^{-1}(x)=(x+5)/3$. Check $f(f^{-1}(x))=x$.

Exam method

1. Write y=f(x). 2. Swap x and y. 3. Solve for y, state a necessary restriction, and verify with composition or reflection.

Use composition as a proof check

After deriving an inverse, compose in one direction: $f(f^{-1}(x))$ should simplify to x on the stated domain. This check is stronger than substituting one convenient number, because it tests the whole algebraic rule. For a non-one-to-one graph, name the branch retained by the restriction, such as $x\geq0$. The restriction is not an afterthought; it is what makes an inverse relation a genuine function.

One-minute retrieval: Inverse Relations and Domain Restrictions

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.