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.