Textbook
If $f:A\to B$ is a bijection, its inverse $f^{-1}:B\to A$ exists. It satisfies $f^{-1}(f(x))=x$ and $f(f^{-1}(y))=y$ for all allowed $x$ and $y$. Every output must be reached and must identify exactly one original input.
For $f(x)=2x$ on $\mathbb R$, the inverse is $f^{-1}(y)=y/2$. This is not $1/f$: the superscript −1 denotes inversion of the function, not a numerical power of its value. Squaring on the entire real line is not invertible: output 4 cannot distinguish inputs 2 and −2. Restricting the domain to $[0,+\infty)$ removes that ambiguity.
Invertibility depends on the domain and codomain as well as the formula. Restricting squaring to nonnegative inputs is insufficient for a bijection into $\mathbb R$: negative outputs remain unreachable. The consistent invertible specification is $f:[0,+\infty)\to[0,+\infty)$, $f(x)=x^2$. The two defining compositions check lossless recovery of inputs and complete recovery across the codomain.
Definitions and further reading: Mathematics for Computer Science.
In plain language
An inverse function reverses the original action without losing information, just as division by two reverses doubling. Every allowed output must identify exactly one original address. Merging two inputs destroys unique recovery, while an unused output has nothing to return. The small minus in $f^{-1}$ is no undo button for lost data.
Example
The illustrative input 3 is doubled and then recovered. The last step compares an inverse function with a reciprocal value; they give different results.
Inverse Functions: Recovering Inputs From Outputs
Forward function
Doubling sends 3 to 6.
Step 1 of 3: Forward function
Read all steps
Forward function
Doubling sends 3 to 6.
- State
- 3current6current
Inverse function
Half of output 6 recovers input 3.
- State
- 6current3current
Not a reciprocal
The reciprocal of f(6) is 1/12, not 3.
- State
- f⁻¹(6): 3current1/f(6): 1/12current
For specific data, both orders give: $f^{-1}(f(3))=f^{-1}(6)=3$ and $f(f^{-1}(6))=f(3)=6$. The first restores the original input; the second restores the original output. For the reciprocal comparison, the calculation first gives $f(6)=12$, followed by $1/12$. The word “inverse” sounds related, but the inputs, operations, and results differ. Explicit parentheses prevent one question from quietly replacing the other.
Related concepts: Preimages.