To find the inverse of the function [tex]\( y = 9x^2 - 4 \)[/tex], we need to follow a series of steps to isolate [tex]\( x \)[/tex] and then solve for [tex]\( y \)[/tex]. Here’s the step-by-step process:
1. Start with the given function:
[tex]\[
y = 9x^2 - 4
\][/tex]
2. To find the inverse, we first swap [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
x = 9y^2 - 4
\][/tex]
3. Solve for [tex]\( y \)[/tex]. Begin by isolating the term containing [tex]\( y \)[/tex]:
[tex]\[
x + 4 = 9y^2
\][/tex]
4. Next, divide both sides of the equation by 9:
[tex]\[
\frac{x + 4}{9} = y^2
\][/tex]
5. To solve for [tex]\( y \)[/tex], take the square root of both sides. Remember to consider the positive and negative square roots:
[tex]\[
y = \pm \sqrt{\frac{x + 4}{9}}
\][/tex]
6. Simplify the expression under the square root:
[tex]\[
y = \pm \sqrt{\frac{x + 4}{9}} = \pm \frac{\sqrt{x + 4}}{3}
\][/tex]
Therefore, the inverse function is:
[tex]\[
y = \frac{ \pm \sqrt{x+4}}{3}
\][/tex]
Comparing this with the given options:
1. [tex]\( y = \frac{\pm \sqrt{x+4}}{9} \)[/tex]
2. [tex]\( y = \pm \sqrt{\frac{x}{9}+4} \)[/tex]
3. [tex]\( y = \frac{\pm \sqrt{x+4}}{3} \)[/tex]
4. [tex]\( y = \frac{\pm \sqrt{x}}{3} + \frac{2}{3} \)[/tex]
The correct answer 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]\[
y = \frac{\pm \sqrt{x+4}}{3}
\][/tex]
