Given #M(x) = (2x)/(x+6)# What is the inverse of #M# and what are the domain and range of the inverse function?

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Answer 1 · 2017-12-12T09:17:48

#M^-1(x)= (6x)/(2-x)#

Domain of #M^-1(x) = (-oo,2)uu(2,+oo)#
Range of #M^-1(x) = (-oo,+oo)#

#M(x) = (2x)/(x+6)#

#x = (2M^-1(x))/(M^-1(x)+6)#

Let #m(x)= M^-1(x)#

#x = (2m(x))/(m(x)+6)#

#xm(x) + 6x = 2m(x)#

#m(x)(2-x) = 6x#

#m(x)= (6x)/(2-x)#

Hence, #M^-1(x)= (6x)/(2-x)#

#M^-1(x)# is defined #forall x in RR# except #x=2#

Hence, the domain of #M^-1(x)# is #(-oo,2)uu(2,+oo)#

Consider:

#lim_(x->2^+) M^-1(x)= -oo#

and

#lim_(x->2^-) M^-1(x)= +oo#

Hence, the range of # M^-1(x)# is #(-oo,+oo)#

We can see these results from the graph of # M^-1(x)# below.

graph{ (6x)/(2-x) [-36.52, 36.54, -18.26, 18.26]}