The function \(\rm f(x)=\log \left(x+\sqrt{x^2 + 1}\right)\) is:

Concept:

• A function f(x) is:
  • Even, if f(-x) = f(x).
  • Odd, if f(-x) = -f(x).
  • Periodic, if f(np ± x) = f(x), for some number p and n ∈ Z.
• log a + log b = log (ab).
• log 1 = 0.

Calculation:

We have \(\rm f(x)=\log \left(x+\sqrt{x^2 + 1}\right)\).

⇒ \(\rm f(-x)=\log \left(-x+\sqrt{x^2 + 1}\right)\).

Now, \(\rm f(x)+f(-x)=\log \left(x+\sqrt{x^2 + 1}\right)+\log \left(-x+\sqrt{x^2 + 1}\right)\)

= \(\rm \log \left[\left(x+\sqrt{x^2 + 1}\right)\left(-x+\sqrt{x^2 + 1}\right)\right]\)

= \(\rm \log \left[\left(\sqrt{x^2 + 1}\right)^2-x^2\right]\)

= \(\rm \log1\)

= 0

⇒ f(-x) = -f(x).

∴ f(x) is an odd function.