Answer:
[tex]\displaystyle y = \pm \sqrt{x + 36}[/tex].
Step-by-step explanation:
To find the inverse of a function, apply the following steps:
- Interchange [tex]x[/tex] and [tex]y[/tex].
- Solve for [tex]y[/tex] in terms of [tex]x[/tex].
For the equation in this question, interchanging [tex]x[/tex] and [tex]y[/tex] yields [tex]x = y^{2} - 36[/tex]. Simplify and solve this equation for [tex]y[/tex]:
[tex]y^{2} = x + 36[/tex].
Before taking the square root of both sides, note that both [tex](-1)^{2}[/tex] and [tex]1^{2}[/tex] are equal to [tex]1[/tex]. In other words, both of the following would evaluate to [tex]x + 36[/tex]:
- [tex]\sqrt{x + 36}[/tex].
- [tex](-1)\, \sqrt{x + 36}[/tex].
Both [tex]y = \sqrt{x + 36}[/tex] and [tex]y = -\sqrt{x + 36}[/tex] are valid inverses of this function. To represent both in one equation:
[tex]\displaystyle y = \pm \sqrt{x + 36}[/tex].
