How do you verify that f(x)=x^2+2, x>=0; g(x)=sqrt(x-2) are inverses?

1 Answer
Feb 13, 2018

Find the inverses of the individual functions.

Explanation:

First we find the inverse of f:

f(x)=x^2+2

To find the the inverse, we interchange x and y since the domain of a function is the co-domain (or range) of the inverse.
f^-1: x = y^2+2
y^2=x-2
y = +-sqrt(x-2)

Since we are told that x>=0,
then it means that f^-1(x)=sqrt(x-2)=g(x)
This implies that g is the inverse of f.

To verify that f is the inverse of g we have to repeat the process for g

g(x)=sqrt(x-2)
g^-1: x=sqrt(y-2)
x^2=y-2
g^-1(x)=x^2-2=f(x)
Hence we have established that f is an inverse of g and g is an inverse of f. Thus the functions are inverses of each other.