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Question

If y=2+2x+1+1x1, then

A
domain of the function is R{1}
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B
domain of the function is R{1,1}
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C
range of the function is R{2}
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D
range of the function is R
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Solution

The correct option is D range of the function is R
y=2+2x+1+1x1
Clearly, function is not defined at x=±1
Hence, domain of the function is R{1,1}

Now, y2=2x+1+1x1
y2=3x1x21 ...(1)
(y2)x23x(y3)=0

For real solutions, Δ0 assuming y20
9+4(y2)(y3)0
4y220y+330
which is always true as Δ<0

Now, put y=2 in equation (1), we get
x=13R{1,1}
Therefore, y=2 also lies in the range.
Hence, range is R.

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