What is the domain and range of 3x-2/5x+1 and the domain and range of inverse of the function?

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Answer 1 · 2016-03-07T22:39:28

Domain is all reals except #-1/5# which is the range of the inverse.
Range is all reals except #3/5# which is the domain of the inverse.

#f(x) = (3x-2)/(5x+1)# is defined and real values for all #x# except #-1/5#, so that is the domain of #f# and the range of #f^-1#

Setting #y=(3x-2)/(5x+1)# and solving for #x# yields

#5xy+y=3x-2#, so

#5xy-3x=-y-2#, and therefore

#(5y-3)x=-y-2#, so, finally

#x=(-y-2)/(5y-3)#.

We see that #y != 3/5#.

So the range of #f# is all reals except #3/5#. This is also the domain of #f^-1#.