Inverse Function - Maple Programming Help

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Inverse Function

Main Concept

Given a function $f\left(x\right)$, the inverse of $f\left(x\right)$ is the function $g\left(x\right)$ which has the property that $y=g\left(x\right)$ exactly when $x=f\left(y\right)$ (for the same $x$ and $y$). That is, the inverse of a function exactly undoes whatever the function does. The inverse of the function $f\left(x\right)$ is commonly denoted by ${f}^{-1}\left(x\right)$.

Note : the inverse of a function is not the same as the reciprocal of that function. For example, the inverse of $f\left(x\right)=\sqrt{x}$ is ${f}^{-1}\left(x\right)={x}^{2}$ (for $x\ge 0)$, not $\frac{1}{\sqrt{x}}$. The notation ${f}^{-1}$ is intended to represent the concept of "inverting the action of $f\left(x\right)$", not "inverting the result of $f\left(x\right)$".

 Algebraically Finding Inverses—Restricting the Domain To find a formula for the inverse of $y=f\left(x\right)$ you switch the roles of $x$ and $y$ and then try to solve for $y$. For example, to find the inverse of the function $y={x}^{3}$ you first switch $x$ and $y$: $x={y}^{3}$. Then you solve for $y$: $\sqrt[3]{x}=y$, or rewriting in standard form, $y=\sqrt[3]{x}$.   Very often it is not possible to carry out the step of solving for $y$, as there may be more than one solution. This means that the original function is not invertible on its natural domain. In such cases, it is usually possible to restrict the domain of the original function to one on which the solve step can be carried to completion.   For example, suppose you want to find the inverse, if it exists, of the function $y={x}^{2}$. Interchanging $x$ and $y$ and trying to solve for $y$ leads you to $y=±\sqrt{x}$, so you can conclude that the original function, $y={x}^{2}$ is not invertible on the entire real line. However, if you restrict the domain to non-negative numbers, so $y={x}^{2}$ when $x\ge 0$, then the inverse will be $y=\sqrt{x}$. Note we could also restrict the domain to be non-positive numbers, so the function is $y={x}^{2}$ when $x\le 0$, in which case the inverse is $y=-\sqrt{x}$. This illustrates that there is invariably some arbitrariness in the choice of restricted domain, and that there can be more than one—sometimes infinitely many—definitions of an "inverse" for a function which is not invertible on its natural domain.

Drag on the graph below to draw an invertible function.

Click Invert to show the process of the function being inverted.

Click Clear to clear the graph and draw again.

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