An inverse function reverses the effect of the original function. If $\mathrm{f}:X\to Y$ is bijective (one-to-one and onto $Y$) and maps $x$ to $y$, then $\mathrm{f}^{-1}:Y\to X$ maps $y$ back to $x$. If $\mathrm{f}(x)=y$, then $\mathrm{f}^{-1}(y)=x$. Thus $\mathrm{f}^{-1}(\mathrm{f}(x))=x$ and $\mathrm{f}(\mathrm{f}^{-1}(y))=y$, wherever the compositions are defined. $\mathrm{f}^{-1}$ exists only when $\mathrm{f}$ is one-one on its stated domain. The graphs of $\mathrm{f}$ and $\mathrm{f}^{-1}$ are reflections in the line $y=x$. $D_{\mathrm{f}}=R_{\mathrm{f}^{-1}}$ and $R_{\mathrm{f}}=D_{\mathrm{f}^{-1}}$. A point $A(r,p)$ on $\mathrm{f}$ corresponds to $A'(p,r)$ on $\mathrm{f}^{-1}$; intercepts are exchanged in the same way. In this example, the graphs meet at $(-1,-1)$ on $y=x$. A function and its inverse do not necessarily meet. Intersections of a function and its inverse need not lie on $y=x$. For example, $\mathrm{f}(x)=-x$ is its own inverse, so their graphs coincide. Solve $\mathrm{f}(x)=\mathrm{f}^{-1}(x)$ directly unless a restriction justifies using $\mathrm{f}(x)=x$. Steps to find the rule of an inverse StepAction ILet $y=\mathrm{f}(x)$. IIMake $x$ the subject. IIIReplace $y$ by $x$ and state the inverse domain and range.

Inverse Functions

An inverse function reverses the effect of the original function. If $\mathrm{f}:X\to Y$ is bijective (one-to-one and onto $Y$) and maps $x$ to $y$, then $\mathrm{f}^{-1}:Y\to X$ maps $y$ back to $x$.

If $\mathrm{f}(x)=y$, then $\mathrm{f}^{-1}(y)=x$. Thus $\mathrm{f}^{-1}(\mathrm{f}(x))=x$ and $\mathrm{f}(\mathrm{f}^{-1}(y))=y$, wherever the compositions are defined.

$\mathrm{f}^{-1}$ exists only when $\mathrm{f}$ is one-one on its stated domain.
The graphs of $\mathrm{f}$ and $\mathrm{f}^{-1}$ are reflections in the line $y=x$.
$D_{\mathrm{f}}=R_{\mathrm{f}^{-1}}$ and $R_{\mathrm{f}}=D_{\mathrm{f}^{-1}}$.
A point $A(r,p)$ on $\mathrm{f}$ corresponds to $A'(p,r)$ on $\mathrm{f}^{-1}$; intercepts are exchanged in the same way.

In this example, the graphs meet at $(-1,-1)$ on $y=x$.

A function and its inverse do not necessarily meet.

Intersections of a function and its inverse need not lie on $y=x$. For example, $\mathrm{f}(x)=-x$ is its own inverse, so their graphs coincide. Solve $\mathrm{f}(x)=\mathrm{f}^{-1}(x)$ directly unless a restriction justifies using $\mathrm{f}(x)=x$.

Steps to find the rule of an inverse

Step	Action
I	Let $y=\mathrm{f}(x)$.
II	Make $x$ the subject.
III	Replace $y$ by $x$ and state the inverse domain and range.

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Step	Action
I	Let \(y=\mathrm{f}(x)\).
II	Make \(x\) the subject.
III	Replace \(y\) by \(x\) and state the inverse domain and range.