Question 1026642
<font face="Times New Roman" size="+2">


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ x^2\ -\ 5]


Substitute *[tex \Large y] for *[tex \Large f(x)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  y\ =\ x^2\ -\ 5]


Solve for *[tex \Large x] in terms of *[tex \Large y]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x^2\ =\ y\ +\ 5]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x\ =\ \pm\sqrt{y\ +\ 5}]


But *[tex \Large x] is constrained to *[tex \Large x\ \geq\ 0] so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  x\ =\ \sqrt{y\ +\ 5}]


Swap positions of the variables:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  y\ =\ sqrt{x\ +\ 5}]


Replace *[tex \Large y] with *[tex \Large f^{-1}(x)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  f^{-1}(x)\ =\ \sqrt{x\ +\ 5}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \text{dom}\left\[f^{-1}(x)\right\]\ =\ \left\{x\ \in\ \mathbb{R}\ |\ x\ \geq\ -5\right\}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \text{dom}\left\[f(x)\right\]\ =\ \left\{x\ \in\ \mathbb{R}\right\}]


*[illustration inversefunction.jpg]


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it

*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  

</font>