Functions can be the inverse of each other if:
$f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)=x$

Meaning that if two functions, 'f', and 'g' exist, and if substitute the results of g(x) into f(x), and the results of f(x) into g(x), the functions are 'inverse' if the results equal the variable value. For example: $\text{If}\phantom{\rule{5px}{0ex}}f\left(x\right)={x}^{2}\phantom{\rule{5px}{0ex}}\text{and}\phantom{\rule{5px}{0ex}}g\left(x\right)=\sqrt{x}$.

We can see what happens if we plug g(x) into f(x), and f(x) into g(x):

${\left(\sqrt{x}\right)}^{2}=\sqrt{{x}^{2}}=x$

After all the substitutions are made, all that is left is 'x', meaning the functions are inverse of each other. The notation for an inverse function is

${f}^{-1}\left(x\right)$

Determining the inverse function is a matter taking the function in question, swapping 'x' and 'y', and solving the equation for 'y'. For example, $\text{If}\phantom{\rule{5px}{0ex}}g\left(x\right)=\sqrt{2x+7}\phantom{\rule{5px}{0ex}}\text{find}\phantom{\rule{5px}{0ex}}{g}^{-1}\left(x\right)$

Meaning that if two functions, 'f', and 'g' exist, and if substitute the results of g(x) into f(x), and the results of f(x) into g(x), the functions are 'inverse' if the results equal the variable value. For example: $\text{If}\phantom{\rule{5px}{0ex}}f\left(x\right)={x}^{2}\phantom{\rule{5px}{0ex}}\text{and}\phantom{\rule{5px}{0ex}}g\left(x\right)=\sqrt{x}$.

We can see what happens if we plug g(x) into f(x), and f(x) into g(x):

${\left(\sqrt{x}\right)}^{2}=\sqrt{{x}^{2}}=x$

After all the substitutions are made, all that is left is 'x', meaning the functions are inverse of each other. The notation for an inverse function is

${f}^{-1}\left(x\right)$

Determining the inverse function is a matter taking the function in question, swapping 'x' and 'y', and solving the equation for 'y'. For example, $\text{If}\phantom{\rule{5px}{0ex}}g\left(x\right)=\sqrt{2x+7}\phantom{\rule{5px}{0ex}}\text{find}\phantom{\rule{5px}{0ex}}{g}^{-1}\left(x\right)$

Evaluating inverse functions.

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