The inverse will be:
![y' = \frac{\sqrt{x+4}}{3}](/tpl/images/0292/3481/9e882.png)
Step-by-step explanation:
In order to find the inverse of the equation, we do a variable change, since we are finding the inverse, :
![f(x) = 9x^{2} - 4](/tpl/images/0292/3481/925cd.png)
![y = 9x^{2} - 4](/tpl/images/0292/3481/c13d6.png)
![x = 9y' ^{2} - 4](/tpl/images/0292/3481/53b86.png)
Now solve for y'.
First add 4 in both sides of the equation and change to the left y'.
![x + 4= 9y'^{2} - 4+4](/tpl/images/0292/3481/dca90.png)
= x + 4
Second divide by 9
/9 = (x + 4)/9
= (x + 4)/9
Now you will have to clear y, with the square root.
" alt="y'^{\frac{2}{2}} = \sqrt{x + 4} / \sqrt{9}" />" /> =
Simplifying terms
![y' = \frac{\sqrt{x+4}}{3}](/tpl/images/0292/3481/9e882.png)
![f^{-1}(x) = \frac{\sqrt{x+4}}{3}](/tpl/images/0292/3481/7af38.png)
You can check the answer by doing the evaluation of the following equation:
(f o
) (x)
substitute the equation for y' or inverse function ![f^{-1}](/tpl/images/0292/3481/11428.png)
f(
)
Now substitue the value into f(x)
You will have
![= 9(\frac{\sqrt{x+4} }{3}} )^{2} - 4\\\\Solving\\\\9(\frac{{x+4} }{9}} ) - 4](/tpl/images/0292/3481/609f6.png)
=x