Concept of Range and Domain

Question

!!e^(sinx)-e^(-sinx)-4=0!!

Then equation has solutions:

a) !!1!!

b) Many solutions

c) !!2!!

d) No solution

Solution

Maximum value of !!e^sinx!! will be !!e= 2.71!!, because !!-1<=sinx<=1!!. So, maximum value of !!e^sinx- e^-sinx!! is less than !!4!!. Therefore no solution exist.

Alternative

Put !!e^(sinx)= t!!

!!=> t^2-4t-1=0!!

!!=> t = (4+-sqrt(20))/2!!

!!=> t= 2+-sqrt5!!

!!t= 2-sqrt5!! is not possible,because !!e^(sinx)!! can't be negative.

!!t= 2+sqrt5=4.23!! also can't be possible. Maximum value of !!sinx!! is !!1!!. Therefore maximum value of !!e^(sinx)!! will be !!e=2.71!!

So, there is no solution exist for given equation.

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