The function f (x) = \(sin\big[\,log\,\big(x+\sqrt{x^2+1}\big)\big]\)

1.	The function f (x) = \(sin\big[\,log\,\big(x+\sqrt{x^2+1}\big)\big]\) is(a) an odd function (b) an even function (c) neither even nor odd (d) None of these.
Answer» Answer : (a) an odd function f(x) = sin \(\big[\,log\,\big(x+\sqrt {x^2+1}\big)\big]\) ∴ f(- x) = sin \(\big[\,log\,\big(-x+\sqrt {(-x)^2+1}\big)\big]\) = \(sin\big[\,log\,(\sqrt{1+x^2}-x)\big]\) = \(sin\big[\,log\,(\sqrt{1+x^2}-x)\frac{(\sqrt{1+x^2}+x)}{(\sqrt{1+x^2}+x)}\big]\) = \(sin\big[\log\,\big(\frac{(1+x^2)-x^2}{\sqrt{1+x^2}+x}\big)\big]\) = \(sin \big[\,log\,(\sqrt{1+x^2}+x)^{-1}\big]\) = \(sin\big[\,-log\,(\sqrt{1+x^2}+x)\big]\) (∴ log a^–1 = – log a) = \(-sin[\,log\,(\sqrt{1+x^2}+x)]\) (∴ sin (– x) = – sin x) = – f (x) ⇒ f is an odd function.

The function f (x) = \(sin\big[\,log\,\big(x+\sqrt{x^2+1}\big)\big]\) is(a) an odd function (b) an even function (c) neither even nor odd (d) None of these.

Answer»

Answer : (a) an odd function

f(x) = sin \(\big[\,log\,\big(x+\sqrt {x^2+1}\big)\big]\)

∴ f(- x) = sin \(\big[\,log\,\big(-x+\sqrt {(-x)^2+1}\big)\big]\)

= \(sin\big[\,log\,(\sqrt{1+x^2}-x)\big]\)

= \(sin\big[\,log\,(\sqrt{1+x^2}-x)\frac{(\sqrt{1+x^2}+x)}{(\sqrt{1+x^2}+x)}\big]\)

= \(sin\big[\log\,\big(\frac{(1+x^2)-x^2}{\sqrt{1+x^2}+x}\big)\big]\)

= \(sin \big[\,log\,(\sqrt{1+x^2}+x)^{-1}\big]\)

= \(sin\big[\,-log\,(\sqrt{1+x^2}+x)\big]\) (∴ log a^–1 = – log a)

= \(-sin[\,log\,(\sqrt{1+x^2}+x)]\) (∴ sin (– x) = – sin x)

= – f (x)

⇒ f is an odd function.