What is the range of #f(x) =x/(abs(x)+1)# ?

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Answer 1 · 2017-11-09T04:09:35

Range: #(-1,+1)#

#f(x) =x/(abs(x)+1)#

First let's find the domain of #f(x)#

#abs(x) >= 0 forall x in RR#

Hence, #(abs(x)+1) >=1 forall x in RR#

#:. f(x)# is defined #forall x in RR#

#-># the domain of #f(x)# is #(-oo, +oo)#

Now let's consider #lim_(x->-oo) f(x) and lim_(x->+oo) f(x) #

#lim_(x->-oo) f(x) = lim_(x->-oo) 1/(abs(x)/x+1/x)#

#= 1/(-1+0) = -1#

#lim_(x->+oo) f(x) = lim_(x->+oo) 1/(abs(x)/x+1/x)#

#=1/(1+0) = +1#

#:.# the range of #f(x)# is #(-1,+1)#

We can deduce this result from the graph of #f(x)# below.

graph{x/(abs(x)+1) [-3.075, 3.081, -1.494, 1.586]}