Inverse functions

A function is a set of input→output pairs. Its inverse is the same set of pairs run backward: if $f(a) = b$, then $f^{-1}(b) = a$. Nothing is computed fresh; every fact about $f^{-1}$ is a fact about $f$ read in the other direction.

That reversal is the master move for tables and graphs. A table of $f$ becomes a table of $f^{-1}$ by swapping the two columns. A graph of $f$ becomes the graph of $f^{-1}$ by reflecting over the line $y = x$, because reflecting over that line is exactly what swapping coordinates does to every point: $(a, b)$ lands on $(b, a)$.

The defining check is composition: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ on the appropriate domains. Do the process, undo the process, and you are back where you started. Any candidate "inverse" that fails this check is not one.

The symbol $f^{-1}$ is historical baggage. For numbers, a superscript $-1$ means reciprocal: $5^{-1} = \tfrac{1}{5}$. For functions, $f^{-1}$ means the inverse function, and it has nothing to do with $\tfrac{1}{f(x)}$.

One numeric check destroys the confusion. Take $f(x) = 2x + 3$, so $f(5) = 13$. The inverse must send 13 back to 5. The reciprocal $\tfrac{1}{f(13)} = \tfrac{1}{29}$ is nowhere near 5. The actual inverse, $f^{-1}(x) = \tfrac{x - 3}{2}$, gives $f^{-1}(13) = 5$ on the nose. When in doubt, run one number through: undoing returns the original input; reciprocating returns a small fraction.

Because the inverse reverses every pair, the inputs of $f^{-1}$ are the outputs of $f$, and vice versa. In one line: the domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.

This is not bookkeeping trivia; it decides what questions even make sense. If $f$ has range $[2, \infty)$, then $f^{-1}(0)$ is meaningless: 0 was never an output of $f$, so it cannot be an input of $f^{-1}$. Whenever you write an inverse, carry the swapped domain and range with it.

Reversing pairs can break the function rule. If $f$ sends two different inputs to the same output, the reversed pairs give that output two different destinations, and $f^{-1}$ is not a function. So an inverse function exists exactly when $f$ is one-to-one: no output repeats. Graphically, every horizontal line crosses the graph at most once.

The fix is domain restriction. $f(x) = x^2$ on all reals fails (both 3 and $-3$ land on 9), but restricted to $x \ge 0$ it is one-to-one and its inverse is $\sqrt{x}$. The restriction is not decoration; without it the "inverse" would have to send 9 to two places at once.

1. Check one-to-one (horizontal line test, or look for repeated outputs).
2. Restrict the domain if needed, keeping the piece the problem cares about.
3. Reverse the pairs: swap table columns, reflect the graph, or solve the rule for the input.
4. Verify by composition and state the swapped domain and range.