How do you find the domain and range of #g(x) = 1/(In x)#?

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Answer 1 · 2017-07-03T10:50:49

Domain: #(0, 1)uu(1, +oo)#
Range: #(-oo, +oo)#

#g(x) = 1/lnx#

#lnx# is defined for #x>0#

#lnx = 0# for #x=1# #-> g(x) # is not defined for #x=1#

Hence, #g(x)# is defined for #x>0, x!=1#

#:.# the domain of #g(x)# is #(0, 1)uu(1, +oo)#

Now consider:

#lim_"(x->1) -"g(x) = -oo#

#lim_"(x->1) +"g(x) = +oo#

Hence, the range of #g(x)# is #(-oo, +oo)#

We can see these results fron the graph of #g(x)#
below.

graph{1/lnx [-10, 10, -5.04, 4.96]}