Inverse Function · aiwiki.page

An inverse function is a function that reverses the input–output correspondence of another function. If $f$ assigns $y$ to $x$ , its inverse, denoted $f^{-1}$ , assigns $x$ to $y$ . This reversal defines a function only when each output under consideration corresponds to exactly one original input. In elementary mathematics, an injective function is therefore described as having an inverse on its range. (openstax.org)

Definition and existence

For a function $f:A\to B$ , an inverse is a function $g:B\to A$ satisfying

g(f(x))=x\quad(x\in A),\qquad f(g(y))=y\quad(y\in B).

Equivalently, using function composition,

g\circ f=\operatorname{id}_A,\qquad f\circ g=\operatorname{id}_B,

where an identity function leaves every element unchanged. A function has such a two-sided inverse precisely when it is bijective: both injective, so distinct inputs have distinct outputs, and surjective, so every element of the codomain occurs as an output. (letsprove.org)

The distinction between codomain and range matters. An injective $f:A\to B$ need not have an inverse defined throughout $B$ , but it does have an inverse $f^{-1}:f(A)\to A$ . Thus the inverse’s domain is the original range, while its range is the original domain. Once these sets and the original function are fixed, the inverse is unique. (openstax.org)

The superscript $-1$ indicates inversion, not a reciprocal:

f^{-1}(x)\ne \frac{1}{f(x)}

in general. For example, if $f(x)=2x+3$ , then $f^{-1}(x)=(x-3)/2$ , whereas its reciprocal is $1/(2x+3)$ . (openstax.org)

Finding and representing inverses

For a function given by a formula, an inverse can often be found by writing $y=f(x)$ , solving this equation for $x$ , and then exchanging the variable names. The resulting expression must be accompanied by its appropriate domain. Checking both composition identities verifies that the proposed function actually reverses the original correspondence. An inverse may exist even when no convenient explicit formula is available. (openstax.org)

For functions of one real variable, inversion has a graphical interpretation. Every ordered pair $(x,y)$ on the original graph becomes $(y,x)$ on the inverse graph. Consequently, the graphs are reflections across $y=x$ . The horizontal line test detects injectivity: no horizontal line may intersect the original graph more than once. If two intersections occur, their exchanged coordinates would give the inverse two outputs for one input. (openstax.org)

A table represents the same operation by exchanging its input and output columns. Repeated original outputs associated with different inputs prevent the reversed table from representing a function. (openstax.org)

Domain restrictions and elementary examples

The polynomial function $f(x)=x^2$ on $\mathbb R$ is not injective because $f(x)=f(-x)$ . Restricting its domain to $[0,\infty)$ produces a bijection onto $[0,\infty)$ , whose inverse is $\sqrt{x}$ . Restricting instead to $(-\infty,0]$ gives the inverse $-\sqrt{x}$ . These are inverses of different restricted functions, not competing inverses of a single fixed bijection. (openstax.org)

Other familiar inverse pairs include the exponential function $e^x:\mathbb R\to(0,\infty)$ and the natural logarithm $\ln x:(0,\infty)\to\mathbb R$ . Periodic trigonometric functions require restrictions: sine on $[-\pi/2,\pi/2]$ is bijective onto $[-1,1]$ , and its inverse is arcsine. The standard inverse trigonometric functions use specified principal intervals to make their values single-valued. (openstax.org)

Continuity and differentiation

A strictly monotonic function on a real interval is injective. If it is also a continuous function, its range is an interval and its inverse is continuous and strictly monotonic in the same direction. The interval property of the range follows from the intermediate value theorem. Merely being nondecreasing or nonincreasing is insufficient for injectivity, since a function may remain constant on a subinterval. (jirka.org)

In calculus, the derivative of an inverse is related to that of the original function. For a continuously differentiable function with nonzero derivative at $a$ , a differentiable local inverse exists near $b=f(a)$ , and

(f^{-1})'(b)=\frac{1}{f'(a)}.

More generally, throughout an interval where the relevant hypotheses hold,

(f^{-1})'(y)=\frac{1}{f'(f^{-1}(y))}.

The chain rule explains this formula by differentiating $f(f^{-1}(y))=y$ . (jirka.org)

A zero derivative does not necessarily prevent invertibility. For example, $x^3$ is bijective on $\mathbb R$ , but its cube-root inverse has no finite derivative at zero. Invertibility and differentiability of the inverse are therefore distinct properties. (openstax.org)

Several variables and local inversion

The inverse function theorem extends local inversion to several variables. If $F:U\to\mathbb R^n$ is continuously differentiable on an open set $U$ , and its Jacobian matrix $DF(a)$ is invertible, suitable neighborhoods of $a$ and $F(a)$ admit a continuously differentiable inverse. Its derivative satisfies

D(F^{-1})(F(a))=[DF(a)]^{-1}.

For a square Jacobian, invertibility is equivalent to a nonzero determinant. This is a local conclusion and does not by itself establish global injectivity. (jirilebl.github.io)

In linear algebra, the special case $F(x)=Mx$ , with $M$ an invertible square matrix, has the global inverse $F^{-1}(y)=M^{-1}y$ . Here the inverse matrix directly represents the inverse function, and the Jacobian is the constant matrix $M$ . (jirilebl.github.io)