Answer:
The inverse will be:
[tex]y' = \frac{\sqrt{x+4}}{3}[/tex]
Step-by-step explanation:
In order to find the inverse of the equation, we do a variable change, since we are finding the inverse, :
[tex]f(x) = 9x^{2} - 4[/tex]
[tex]y = 9x^{2} - 4[/tex]
[tex]x = 9y' ^{2} - 4[/tex]
Now solve for y'.
First add 4 in both sides of the equation and change to the left y'.
[tex]x + 4= 9y'^{2} - 4+4[/tex]
[tex]9y'^{2}[/tex] = x + 4
Second divide by 9
[tex]9y'^{2}[/tex]/9 = (x + 4)/9
[tex]y'^{2}[/tex] = (x + 4)/9
Now you will have to clear y, with the square root.
[tex][tex]y'^{\frac{2}{2}} = \sqrt{x + 4} / \sqrt{9}[/tex][/tex] =
Simplifying terms
[tex]y' = \frac{\sqrt{x+4}}{3}[/tex]
[tex]f^{-1}(x) = \frac{\sqrt{x+4}}{3}[/tex]
You can check the answer by doing the evaluation of the following equation:
(f o [tex]f^{-1}[/tex] ) (x)
substitute the equation for y' or inverse function [tex]f^{-1}[/tex]
f([tex]\frac{\sqrt{x+4} }{3}[/tex])
Now substitue the value into f(x)
You will have
[tex]= 9(\frac{\sqrt{x+4} }{3}} )^{2} - 4\\\\Solving\\\\9(\frac{{x+4} }{9}} ) - 4[/tex]
=x
