Question

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

Answer 1

Answer:-

The two inverses are,

`y = \sqrt {(x+4)/(9)}`

and,
`y =  - \sqrt {(x+4)/(9)}`

Step-by-step calculation:-

`y = 9 * x^(2) - 4`

⇒
`x^(2) = (y+4)/(9)`

⇒
`x = \mp \sqrt {(y+4)/(9)}`

So, the two inverses are,

`y = \sqrt {(x+4)/(9)}`

and,
`y =  - \sqrt {(x+4)/(9)}`

Answer 2

`f^(-1)(x)=\pm (√(x+4) )/(3)`

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:

`(y+4)/(9)=(9x^2)/(9)  \\(y+4)/(9)=x^2`

Square root from both sides:

`√(x^2)=\pm \sqrt{(y+4)/(9)}  \\x=\pm (√(y+4) )/(3)`

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

`y=\pm (√(x+4) )/(3)`

So:

`f^(-1)(x)=\pm (√(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((√(x+4) )/(3))^2 -4=x+4-4=x`

`(f^(-1)\hspace{3}o\hspace{3}f)(x)=x=(√(9x^2-4+4) )/(3) =(√(9x^2) )/(3) =(3x)/(3) =x`