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