How do you verify that #f(x)=3x+5; g(x)=1/3x-5/3# are inverses?

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Answer 1 · 2017-02-22T05:47:06

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

There are three ways to verify that these functions are inverse functions:

#(f @ g)(x) = f(g(x)) = 3(1/3x-5/3)+5 = x-5+5 = x# #(g @ f)(x) = g(f(x)) = 1/3(3x+5)-5/3 = x + 5/3 - 5/3 = x# Therefore #f(x) and g(x)# are inverse functions.
Let #f(x) = y#. Interchange #x# with #y#, then solve for #y#. You should get #g(x)#: #x = 3y+5#, #x-5=3y#, #y=1/3x - 5/3 = g(x)#
Graph both functions and the #y=x# line. The functions should be reflections of each other across the #y=x# line. This means for each point #(a,b)# from #f(x)# there should be a #(b,a)# point on #g(x)#.