Question

Identify the inverse function of f(x) = VX - 2 + 3.

Answer 1

`\boxed{ {f}^( - 1) (x) = {(x - 3)}^(2) + 2}`

Option D is the correct option

Explanation:

`\mathsf{f(x) = √(x - 2) + 3}`

Replace f(x) with y

`\mathsf{y = √(x - 2) + 3}`

Interchange variables

`\mathsf{x = √(y - 2) + 3}`

`\mathsf{{(x - 3)}^(2) = {( √(y - 2)) }^(2) }`

`\mathsf{ {(x - 3)}^(2) = y - 2}`

`\mathsf{ y = {(x - 3)}^(2) + 2}`

Replace y with f ⁻¹( x )

`\mathsf{ {f}^( - 1) (x) = {(x - 3)}^(2) + 2}`

Hope I helped!

Best regards!

Answer 2

`\huge\boxed{f^(-1)(x) = (x-3)^2+2}`

Explanation:

`f(x) = √(x-2) + 3`

Replace y = f(x)

`y = √(x-2) + 3`

Exchange x and y

`x = √(y-2)+3`

Solve for y

`x = √(y-2)+3`

Subtracting both sides by 3

`x - 3 = √(y-2)`

Taking square on both sides

`(x-3)^2 = y -2`

Adding 2 to both sides

`y = (x-3)^2+2`

Substitute y =
`f^(-1)(x)`

`f^(-1)(x) = (x-3)^2+2`