How do you show that #f(x)=1-x^3# and #g(x)=root3(1-x)# are inverse functions algebraically and graphically?

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Answer 1 · 2017-06-06T20:24:51

See the explanation below

If a function #f(x)# has an inverse #f^-1(x)# then

The composition

#f(f^-1(x))=x#

and #f^-1(f(x))=x#

Here, we have

#f(x)=1-x^3#

#g(x)=root(3)(1-x)#

Then

#f(g(x))=f(root(3)(1-x))=1-(root(3)(1-x))^3=1-(1-x)=x#

#g(f(x))=g(1-x^3)=root(3)((1-(1-x^3)))=root(3)(x^3)=x#

Therefore,
algebraically #f(x)# and #g(x)#are inverse functions

graph{(y-1+x^3)(y-root(3)(1-x))(y-x)=0 [-3.117, 3.813, -0.884, 2.58]}

Graphically, the functions #f(x)# and #g(x)# are symmetric with

respect to the line #y=x#. So they are inverse functions.