The function f(x) = (x+x^2+1) is:

Question

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

Accepted Answer

$$f\left( x \right) = \log \left( {x + \sqrt {{x^2} + 1} } \right)$$

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

$$ = \log \left\{ {{{ - {x^2} + {x^2} + 1} \over {x + \sqrt {{x^2} + 1} }}} \right\}$$

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

$$ \Rightarrow f\left( x \right)\,\,\,\,$$ is an odd function.

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