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Question

Find the domain of definition of the following function:
f(x)=log0.5(x2+x+6)+1x2+2x

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Solution

Since, log0.5(x2+x+6)0
As base is less than 1
0<x2+x+61
Also for 1x2+2x

x2+2x0x(x+2)0x0,2....(i)

Now,
x2+x+6>0x2x6<0(x3)(x+2)<0x(2,3) and x2+x+61x2+x+50x2x50x[1212,1+212]...(2)

Taking intersection from (i) and (ii) , we get
x[1212,1+212]{0,2}

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