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UPSEE Solved Model Paper 7

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Question : 121 of 150

Marks: +1, -0

The range of the function

x^2+\frac{1}{x^2+1}

is

[1 , ∞)
[2 , ∞)
[3/2 , ∞)
None of these

Solution:

Given

f (x) =

x^2+\frac{1}{x^2+1}

We have to find the range of f (x)

Now,

f (x) =

x^2+\frac{1}{x^2+1}

=

1+x^2-1

+

\frac{1}{x^2+1}

[Adding and subtracting 1]

=

(x^2+1)

+

\left(\frac{1}{x^2+1}-1\right)

=

1+x^2-\frac{x^2}{x^2+1}

=

1+x^2\left(1-\frac{1}{1+x^2}\right)

≥ 1 for all x ∊ R

[Because

x^2

≥ 0 and 0 <

\frac{1}{1+x^2}

≤ 1 for all x ∊ R]

So, range of f = [1 , ∞) for all x ∊ R

Hence, option 'A' is correct.

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