To find the inverse of the function \( y = 9x^2 - 4 \), we need to follow these steps:
1. Express \( y \) as a function of \( x \):
[tex]\[ y = 9x^2 - 4 \][/tex]
2. Replace \( y \) with \( x \) and \( x \) with \( y \) to find the inverse function:
[tex]\[ x = 9y^2 - 4 \][/tex]
3. Solve this equation for \( y \).
Start with:
[tex]\[ x = 9y^2 - 4 \][/tex]
Isolate the term with \( y^2 \):
[tex]\[ x + 4 = 9y^2 \][/tex]
Divide both sides by 9:
[tex]\[ \frac{x + 4}{9} = y^2 \][/tex]
Take the square root of both sides:
[tex]\[ y = \pm \sqrt{\frac{x + 4}{9}} \][/tex]
Since \(\sqrt{\frac{x + 4}{9}} = \frac{\sqrt{x + 4}}{\sqrt{9}}\):
[tex]\[ y = \pm \frac{\sqrt{x + 4}}{3} \][/tex]
Therefore, the inverse of the function \( y = 9x^2 - 4 \) is:
[tex]\[ y = \pm \frac{\sqrt{x + 4}}{3} \][/tex]
Hence, the correct equation is:
[tex]\[ y = \frac{\pm \sqrt{x+4}}{3} \][/tex]
The correct choice is the third one:
[tex]\[ y = \frac{\pm \sqrt{x + 4}}{3} \][/tex]
