Which equation is the inverse of [tex]$y=9x^2-4$[/tex]?

A. [tex]y=\frac{\pm \sqrt{x+4}}{9}[/tex]
B. [tex]y=\pm \sqrt{\frac{x}{9}+4}[/tex]
C. [tex]y=\frac{\pm \sqrt{x+4}}{3}[/tex]
D. [tex]y=\frac{\pm \sqrt{x}}{3}+\frac{2}{3}[/tex]



Answer :

To find the inverse of the function [tex]\( y = 9x^2 - 4 \)[/tex], we will follow these steps:

1. Express the original function in terms of [tex]\( y \)[/tex] and solve for [tex]\( x \)[/tex]:

[tex]\[ y = 9x^2 - 4 \][/tex]

2. Add 4 to both sides to isolate the [tex]\( x^2 \)[/tex] term on one side:

[tex]\[ y + 4 = 9x^2 \][/tex]

3. Divide both sides of the equation by 9 to isolate [tex]\( x^2 \)[/tex]:

[tex]\[ \frac{y + 4}{9} = x^2 \][/tex]

4. Take the square root of both sides to solve for [tex]\( x \)[/tex]:

[tex]\[ x = \pm \sqrt{\frac{y + 4}{9}} \][/tex]

5. Simplify the square root expression:

[tex]\[ x = \pm \frac{\sqrt{y + 4}}{3} \][/tex]

Now, to write the inverse function, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:

[tex]\[ y = \pm \frac{\sqrt{x + 4}}{3} \][/tex]

Thus, the correct inverse function equation is:

[tex]\[ y = \frac{\pm \sqrt{x + 4}}{3} \][/tex]

Therefore, the answer is:
[tex]\[ \boxed{\frac{ \pm \sqrt{x+4}}{3}} \][/tex]