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]
