To determine the inverse of the equation [tex]\( y = 9x^2 - 4 \)[/tex], we need to follow a series of steps to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
1. Start with the original equation:
[tex]\[ y = 9x^2 - 4 \][/tex]
2. Isolate the term with [tex]\( x \)[/tex]:
[tex]\[ y + 4 = 9x^2 \][/tex]
3. Solve for [tex]\( x^2 \)[/tex] by dividing both sides by 9:
[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 expression:
[tex]\[ x = \pm \frac{\sqrt{y + 4}}{3} \][/tex]
Now we have the inverse of the function in terms of [tex]\( y \)[/tex]:
[tex]\[ x = \pm \frac{\sqrt{y + 4}}{3} \][/tex]
To express [tex]\( x \)[/tex] in the form of [tex]\( y \)[/tex], we rewrite the equation by swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ y = \pm \frac{\sqrt{x + 4}}{3} \][/tex]
Among the given options:
- [tex]\( y = \frac{\pm \sqrt{x+4}}{9} \)[/tex]
- [tex]\( y = \pm \sqrt{\frac{x}{9} + 4} \)[/tex]
- [tex]\( y = \frac{\pm \sqrt{x+4}}{3} \)[/tex]
- [tex]\( y = \frac{\pm \sqrt{x}}{3} + \frac{2}{3} \)[/tex]
The correct inverse function that we derived is:
[tex]\[ y = \frac{\pm \sqrt{x+4}}{3} \][/tex]
Hence, the equation that represents the inverse of [tex]\( y = 9x^2 - 4 \)[/tex] is:
[tex]\[ \boxed{y = \frac{\pm \sqrt{x+4}}{3}} \][/tex]
