The range of the function f (x) = x2 +\(\frac{1}{x^2+1}\) (a) [1, ∞

1.	The range of the function f (x) = x2 +\(\frac{1}{x^2+1}\) (a) [1, ∞) (b) [2, ∞) (c) [ \(\frac{3}{2}\), ∞ ) (d) None of these
Answer» Answer: (a) [1, ∞) Given f(x) = x² + \(\frac{1}{x^2+1}\) = \(\frac{x^4+x^2+1}{x^2+1}\) = \(\frac{x^4+2x^2+1-x^2}{x^2+1} = \frac{(x^2+1)^2-x^2}{x^2+1}\) = \((x^2+1) - \frac{x^2}{x^2+1}\) = 1+ x² \(\big(1-\frac{1}{x^2+1}\big)\) \(\geq\) 1 \(\forall\) x \(\in\) R ∴ Range of f (x) = [1, ∞ ).

The range of the function f (x) = x2 +\(\frac{1}{x^2+1}\) (a) [1, ∞) (b) [2, ∞) (c) [ \(\frac{3}{2}\), ∞ ) (d) None of these

Answer»

Answer: (a) [1, ∞)

Given

f(x) = x² + \(\frac{1}{x^2+1}\) = \(\frac{x^4+x^2+1}{x^2+1}\)

= \(\frac{x^4+2x^2+1-x^2}{x^2+1} = \frac{(x^2+1)^2-x^2}{x^2+1}\)

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

= 1+ x² \(\big(1-\frac{1}{x^2+1}\big)\) \(\geq\) 1 \(\forall\) x \(\in\) R

∴ Range of f (x) = [1, ∞ ).