Question

Find the inverse of f(x) = l, x > 2. Determine whether the inverse is also a function, and find the domain and range of the inverse.

Answer 1

The given function is,

`f(x)=-\sqrt[]{x-2}\text{ , x>2}`

Steps to find the inverse:

Step 1

Replace f(x) with y.

`y=-\sqrt[]{x-2}\text{ }`

Step 2

Replace x with y and y with x in equation from step 1.

`x=-\sqrt[]{y-2}`

Step 3

Solve the equation from step 2 for y.

`\begin{gathered} x^2=y-2 \\ x^2+2=y \\ y=x^2+2 \end{gathered}`

Step 4

Replace y with f^-1(x).

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

Therefore, the inverse of f(x) is ,

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

The inverse function f(x) is in the shape of a parabola opening upwards.

The graph of inverse function is,

If a vertical line drawn does not intersect tha graph more than once, then the graph is of a function(vertical line test).

Since a vertical line does not intersect the graph more than once, the inverse of f(x) is a function.

Since the inverse of f(x) is defined at all points in the interval (-∞, ∞), the domain of the inverse of f(x) is (-∞, ∞).

The range of the inverse of f(x) is [2,∞)