How do you verify if #f(x)=x^2+5; x>=0; g(x)=sqrt(x-5)# are inverse functions?

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Answer 1 · 2017-02-18T07:10:45

If #(f @ g)(x) = x# and #(g @ f)(x) = x# then they are inverse functions

See if #(f @ g)(x) = x# and #(g @ f)(x) = x#:
#(f @ g)(x) = (sqrt(x-5))^2 + 5 = x-5+5 = x#
#(g @ f)(x) =sqrt(x^2+5-5) = sqrt(x^2) = x#
Therefore #f(x)# and #g(x)# are inverse functions.
You can also start with one of the functions, let #f(x) = y#, switch the #x# and #y# variables and solve for the #y#:
#y = x^2 +5#
Switch #x# and #y#: #x = y^2 + 5#
Solve for #y#: #x-5 = y^2; y = sqrt(x-5) = g(x)#
Therefore #f(x)# and #g(x)# are inverse functions.
You can graph the two functions and the #y=x# function. If the two functions are inverse functions, they will reflect each other along the #y=x# line. Make sure when you look at the graphs, you take into account any domain restrictions (e.g. #x >=0#).