The correct option is D range of the function is R
y=2+2x+1+1x−1
Clearly, function is not defined at x=±1
Hence, domain of the function is R−{−1,1}
Now, y−2=2x+1+1x−1
⇒y−2=3x−1x2−1 ...(1)
⇒(y−2)x2−3x−(y−3)=0
For real solutions, Δ≥0 assuming y−2≠0
⇒9+4(y−2)(y−3)≥0
⇒4y2−20y+33≥0
which is always true as Δ<0
Now, put y=2 in equation (1), we get
x=13∈R−{−1,1}
Therefore, y=2 also lies in the range.
Hence, range is R.