Left and Right Inverses¶

Definition¶

Let $f\colon A\to B$ be a function. A left inverse of $f$ is a function $\ell\colon B\to A$ such that

A right inverse of $f$ is a function $r\colon B\to A$ such that

Interpretation¶

A left inverse undoes the function after the function is applied. A right inverse gives a way to hit every target value.

The left inverse sends each image point back to its original input; it can take arbitrary values on unused points of $B$. The right inverse chooses one preimage for every point of $B$.

The equations use composition: applying $\ell$ after $f$ must recover the identity function on the input set for a left inverse. A right inverse instead makes applying $f$ after $r$ recover the identity on the output set.

Key Results¶

• A function has a left inverse only if it is injective.
• A function has a right inverse only if it is surjective. Conversely, a surjective function has a right inverse, obtained by choosing one preimage for every output.
• A two-sided inverse is both a left inverse and a right inverse.

Examples¶

• The inclusion $\boldsymbol{i}\colon\mathbb{R}\to\mathbb{R}^2$${\mathbb{R}}^2$, $\boldsymbol{i}(t)=(t,0)$$(t,0)$, has a left inverse $\ell(x,y)=x$$x$.
• The projection $p\colon\mathbb{R}^2\to\mathbb{R}$$\mathbb{R}$, $p(x,y)=x$$x$, has a right inverse $\boldsymbol{r}(t)=(t,0)$$(t,0)$.

Prerequisites¶

• Composition of Functions
• Identity Function

Related Concepts¶

• Injective Function
• Surjective Function
• Bijection and Inverse Function

Sources¶

• Math IV: Multivariate Calculus · Week 3, Lecture 1
• Math V: Advanced Linear Algebra · Week 7 (revisited)

Aliases: one-sided inverse Keywords: left inverse, right inverse Symbols: real numbers

Last updated: 20 October 2026 03:02