Inverse Functions - Maple Help
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Inverse Functions

 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 values of $x$ 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 ${f}^{-1}\left(x\right)$.   Some functions have inverses (they are called invertible) and some do not (they are called non-invertible). An easy way to tell if a function is invertible or not is whether or not it passes the horizontal line test:   If any horizontal line passes through more than one point on the graph of y=f(x), the function f(x) is not invertible. If no such horizontal line exists, the function is invertible.   Even if  $f\left(x\right)$ is not invertible, it might still have a partial inverse. If you restrict the domain of $f\left(x\right)$, creating a new function $g\left(x\right)$ which does pass the horizontal line test, then $g\left(x\right)$ is invertible, and its inverse is called a partial inverse of  $f\left(x\right)$.   In order to graph a function's inverse, simply reflect its graph through the line .   Note : Even though they look similar, the inverse of $f\left(x\right)$, denoted ${f}^{-1}\left(x\right)$, is not the same as the reciprocal of $f\left(x\right)$, which can be written $f{\left(x\right)}^{-1}$. For example, the inverse of the function  $f\left(x\right)={x}^{2}$ is the function ${f}^{-1}\left(x\right)=\sqrt{x}$ , not the expression $\frac{1}{\sqrt{x}}$. The notation  ${f}^{-1}$ is intended to represent the concept of "inverting the action of $f\left(x\right)$", not "inverting the result of $f\left(x\right)$".

Choose an example function from the drop down menu or define your own function by typing in the box. Restrict its domain so that it becomes an invertible function, and plot the inverse. Note that there are often many ways to restrict a function's domain so that it becomes invertible, only one of which is provided here.

Example =

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