How do you use composition of functions to show that #f(x)=(2+x)/x# and #f^-1(x) = 2/(x-1)# are inverses?

1 Answer
Karthik G
Jan 13, 2016

Step by step explanation is given below.

Explanation:

To prove #f(x)# and #f^-1(x)# we need to show
#(fcirc f^-1)(x)=x# and #(f^-1circf)(x)=x#

#f(x)=(2+x)/x# and #f^-1(x) = 2/(x-1)#

#(fcircf^-1)(x)#

# = (2+2/(x-1))/(2/(x-1))#

# =((2(x-1))/(x-1)+2/(x-1))/(2/(x-1))#

#=(2(x-1)+2)/(x-1) xx (x-1)/2#

#=(2x-2+2)/(x-1)xx(x-1)/2#

#=(2x)/(x-1)xx(x-1)/2#

#=(cancel(2)x)/cancel(x-1)xxcancel(x-1)/cancel(2)#

#=x#

We got that #(fcirc f^-1)(x)=x#

Similarly, we can prove #(f^-1circf)(x)=x#
#(f^-1circf)(x)#

#=2/((2+x)/x - 1)#

#=2/((2+x)/x- x/x)#

#=2/((2+x-x)/x)#

#=2/((2+cancel(x)-cancel(x))/x)#

#=2/(2/x)#

#=2/1 xx x/2#

#=cancel(2)/1xxx/cancel(2)#

#=x#

Therefore, #(f^-1circf)(x)=x#

Thus were are able to show that the functions are inverses of each other.

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