The given function is,
[tex]f(x)=-\sqrt[]{x-2}\text{ , x>2}[/tex]
Steps to find the inverse:
Step 1
Replace f(x) with y.
[tex]y=-\sqrt[]{x-2}\text{ }[/tex]
Step 2
Replace x with y and y with x in equation from step 1.
[tex]x=-\sqrt[]{y-2}[/tex]
Step 3
Solve the equation from step 2 for y.
[tex]\begin{gathered} x^2=y-2 \\ x^2+2=y \\ y=x^2+2 \end{gathered}[/tex]
Step 4
Replace y with f^-1(x).
[tex]f^{-1}(x)=x^2+2[/tex]
Therefore, the inverse of f(x) is ,
[tex]f^{-1}(x)=x^2+2[/tex]
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,∞)
