Question
The range of the function $$f\left( x \right) = {x^2} + \frac{1}{{{x^2} + 1}}$$ is :
A.
$$\left[ {1,\, + \infty } \right)$$
B.
$$\left[ {2,\, + \infty } \right)$$
C.
$$\left[ {\frac{3}{2},\, + \infty } \right)$$
D.
none of these
Answer :
$$\left[ {1,\, + \infty } \right)$$
Solution :
$$\eqalign{
& f\left( x \right) = {x^2} + \frac{1}{{{x^2} + 1}} - 1 + 1 = 1 + {x^2} - \frac{{{x^2}}}{{1 + {x^2}}} \cr
& = 1 + {x^2}\left( {1 - \frac{1}{{1 + {x^2}}}} \right) \geqslant 1{\text{ for all }}x\, \in \,R \cr} $$
The domain $$f=R.$$ Clearly, as $$x$$ increases $$f\left( x \right)$$ increases.