What is the domain and range of #cos(1/x)#?

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Answer 1 · 2018-02-28T04:53:33

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

#f(x) = cos(1/x)#

f(x) is defined #forall x in RR: x!=0#

Hence, the domain of #f(x)# is #(-oo,0)uu(0,+oo)#

Consider, #-1<=f(x)<= +1 forall x in RR: x!=0#

And, #lim_(x->-oo) f(x) =1#

And, #lim_(x->+oo) f(x) =1#

Also, #lim_(x->0) f(x)# does not exist.

Hence, the range of #f(x)# is #[-1,+1]#

We may deduce these results from the graph of #fx)# below.

graph{cos(1/x) [-3.464, 3.465, -1.734, 1.728]}