which equation is the inverse of $y = 9x^2 - 4$?\n$\\bigcirc\\ y = \\frac{\\pm\\sqrt{x + 4}}{9}$\n$\\bigcirc\…

Explanation:

Step1: Swap x and y

To find the inverse of a function, we first swap the roles of ( x ) and ( y ) in the equation ( y = 9x^2 - 4 ). So we get ( x = 9y^2 - 4 ).

Step2: Solve for y

First, we isolate the term with ( y^2 ). Add 4 to both sides of the equation: ( x + 4 = 9y^2 ). Then, divide both sides by 9: ( \frac{x + 4}{9}=y^2 ). Now, take the square root of both sides. Remember that when taking the square root, we have to consider both the positive and negative roots, so ( y=\pm\sqrt{\frac{x + 4}{9}} ). We can simplify ( \sqrt{\frac{x + 4}{9}} ) as ( \frac{\sqrt{x + 4}}{3} ) (since ( \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}} ) for non - negative ( a ) and positive ( b )). So the inverse function is ( y = \pm\frac{\sqrt{x + 4}}{3} ).

Answer:

( y=\pm\frac{\sqrt{x + 4}}{3} ) (corresponding to the third option: ( y=\frac{\pm\sqrt{x + 4}}{3} ))