Question

Which equation is the inverse of y = 9x2 – 4?

Answer 1

Answer:-

The two inverses are,

$y = \sqrt {\frac{x+4}{9}}$

and,  $y = - \sqrt {\frac{x+4}{9}}$

Step-by-step calculation:-

$y = 9 \times x^{2} - 4$

⇒$x^{2} = \frac{y+4}{9}$

⇒$x = \mp \sqrt {\frac{y+4}{9}}$

So, the two inverses are,

$y = \sqrt {\frac{x+4}{9}}$

and,  $y = - \sqrt {\frac{x+4}{9}}$

Answer 2

Answer:

$f^{-1}(x)=\pm \frac{\sqrt{x+4} }{3}$

Step-by-step explanation:

In order to find the inverse of $y=9x^2-4$ we need to follow the next steps:

Step 1: Solve for x

Add 4 to both sides:

$y+4=9x^2-4+4\\y+4=9x^2$

Divide by 9 from both sides:

$\frac{y+4}{9}=\frac{9x^2}{9} \\\frac{y+4}{9}=x^2$

Square root from both sides:

$\sqrt{x^2}=\pm \sqrt{\frac{y+4}{9}} \\x=\pm \frac{\sqrt{y+4} }{3}$

Step 2: Replace every x with a y and replace every y with an x.

$y=\pm \frac{\sqrt{x+4} }{3}$

So:

$f^{-1}(x)=\pm \frac{\sqrt{x+4} }{3}$

Step 3: Verify your work by checking that:

$(f \hspace{3}o\hspace{3}f^{-1})(x)=x\\(f^{-1}\hspace{3}o\hspace{3}f)(x)=x$

$(f \hspace{3}o\hspace{3}f^{-1})(x)=x=9(\frac{\sqrt{x+4} }{3})^2 -4=x+4-4=x$

$(f^{-1}\hspace{3}o\hspace{3}f)(x)=x=\frac{\sqrt{9x^2-4+4} }{3} =\frac{\sqrt{9x^2} }{3} =\frac{3x}{3} =x$