To determine the inverse of the function [tex]\( y = 9x^2 - 4 \)[/tex], we need to proceed through a series of steps to express it as [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], and then solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
1. Start with the original equation:
[tex]\[
y = 9x^2 - 4
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
- First, isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
y + 4 = 9x^2
\][/tex]
- Next, divide both sides by 9:
[tex]\[
\frac{y + 4}{9} = x^2
\][/tex]
- Then, take the square root of both sides. Remember that solving for [tex]\( x \)[/tex] involves both the positive and negative square roots:
[tex]\[
x = \pm \sqrt{\frac{y + 4}{9}}
\][/tex]
3. Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- To write [tex]\( x \)[/tex] explicitly, we get:
[tex]\[
x = \frac{\pm \sqrt{y + 4}}{3}
\][/tex]
4. Form the inverse function:
- To find the inverse function, switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
y = \frac{\pm \sqrt{x + 4}}{3}
\][/tex]
Thus, the inverse of the function [tex]\( y = 9x^2 - 4 \)[/tex] is given by:
[tex]\[
\boxed{y = \frac{\pm \sqrt{x + 4}}{3}}
\][/tex]
Checking the options provided:
- First Option: [tex]\( y = \frac{\pm \sqrt{x + 4}}{9} \)[/tex]
- Second Option: [tex]\( y = \pm \sqrt{\frac{x}{9} + 4} \)[/tex]
- Third Option: [tex]\( y = \frac{\pm \sqrt{x + 4}}{3} \)[/tex]
- Fourth Option: [tex]\( y = \frac{\pm \sqrt{x}}{3} + \frac{2}{3} \)[/tex]
Clearly, the correct answer is the third option:
[tex]\[
\boxed{y = \frac{\pm \sqrt{x + 4}}{3}}
\][/tex]
