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# Inverse function

The inverse function of a bijection $f:X\to Y$ is the function $f^{-1}:Y\to X$ that maps each value to its unique pre-image $$y=f(x) \Longleftrightarrow x=f^{-1}(y).$$

The inverse of $x^2$ from $[0, +\infty)$ to $[0, +\infty)$ is $\sqrt{x}$. This is because $y=x^2$ is equivalent to $x = \sqrt{y}$. The inverse of $\ln x$ is $e^x$.

The domain of the inverse $f^{-1}$ is the range of the original function $f$. The range of the inverse $f^{-1}$ is the domain of $f$. To summarise, $$D_{f^{-1}} = R_f,\quad R_{f^{-1}} = D_f.$$

$(0,+\infty)$ is the range of $e^x$ and the domain of $\ln x$.

The inverse function is bijective and its inverse is the original function $$(f^{-1})^{-1} = f.$$

The graph of the inverse function is deduced from the graph of the function by a reflection across the line $y=x$. This is because the two graphs simply have the $x$ and $y$ axes inverted.

Graph of $f^{-1}$ and graph of $f$