Explanation

(a)
Given, $f(x) = x^{2} + \frac{1}{x^{2} + 1}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \left( x^{2} + 1 \right) - \left( \frac{x^{2}}{x^{2} + 1} \right)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 1 + x^{2}\left( 1 - \frac{1}{x^{2} + 1} \right) \geq 1,\ \forall\ x \in R$
Hence, range of $f(x)$ is $\lbrack 1,\ \infty)$.