UPSEE Solved Model Paper 7

Given
f (x) = x2+

1

x2+1

We have to find the range of f (x)
Now,
f (x) = x2+

1

x2+1

= 1+x2−1 +

1

x2+1

[Adding and subtracting 1]
= (x2+1) + (

1

x2+1

−1) = 1+x2−

x2

x2+1

= 1+x2(1−

1

1+x2

) ≥ 1 for all x ∊ R
[Because x2 ≥ 0 and 0 <

1

1+x2

≤ 1 for all x ∊ R]
So, range of f = [1 , ∞) for all x ∊ R
Hence, option 'A' is correct.