How do you show that #f(x)=(x-1)/(x+5)# and #g(x)=-(5x+1)/(x-1)# are inverse functions algebraically and graphically?

1 Answer
Aug 30, 2017

See below.

Explanation:

Two functions #f(x)# and #g(x)# are inverses of each other if and only if #f(g(x)) = g(f(x)) = x#.

#f(g(x))= f(-(5x+1)/(x-1)) = (-(5x+1)/(x-1) -1) / (-(5x+1)/(x-1) +5)#

#=(-(5x+1)/(x-1) - (x-1)/(x-1))/(-(5x+1)/(x-1) + (5(x-1))/(x-1))#

#=((-5x-1-x+1)/(x-1))/((-5x-1+5x-5)/(x-1))#

#=(-5x-1-x+1)/(-5x-1+5x-5)#

#=(-6x)/-6#

#=x#

#g(f(x))= g((x-1)/(x+5))=-(5((x-1)/(x+5))+1)/((x-1)/(x+5)-1)#

#=-((5x-5)/(x+5) + (x+5)/(x+5))/((x-1)/(x+5)-(x+5)/(x+5))#

#=-((5x-5+x+5)/(x+5))/((x-1-x-5)/(x+5))#

#=-(5x-5+x+5)/(x-1-x-5)#

#=-(6x)/(-6)#

#=x#

Since we have proved #f(g(x))=x# and #g(f(x))=x#, we can say #f(g(x))=g(f(x))=x#.

#color(white)I#

In the graph below, #f(x)# is shown in red and #g(x)# in blue. If two functions #f(x)# and #g(x)# are inverses, then they will be reflections of each other over the line #y=x# (shown in green).

desmos.com

It's clear to see that #f(x)# and #g(x)# are reflections of each other over the line!