How do you show that #f(x)=9-x^2# and #g(x)=sqrt(9-x)# are inverse functions algebraically and graphically?

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Answer 1 · 2017-01-31T17:54:30

See proof below

We calculate the composition of the functions

#f(x)=9-x^2#

#g(x)=sqrt(9-x)#

#f(g(x))=f(sqrt(9-x))=9-(sqrt(9-x))^2#

#=9-(9-x)=x#

#g(f(x))=g(9-x^2)=sqrt(9-(9-x^2))#

#sqrtx^2=x#

Therefore, #f(x)# and #g(x)# are inverses

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

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

Graphically, #f(x)# and #g(x)# are symmetrics wrt #y=x#

graph{(y-9+x^2)(y-sqrt(9-x))(y-x)=0 [-1.78, 43.84, -9.37, 13.44]}