Why is $\log(\sqrt{x^2+1}+x)$ odd?

$$f(x) = \log(\sqrt{x^2+1}+x)$$ I can't figure out, why this function is odd. I mean, of course, its graph shows, it's odd, but when I investigated $f(-x)$, I couldn't find way to $-\log(\sqrt{x^2+1}+x)$.

• $f(x) = \log(\sqrt{x^2+1}+x) = \text{ArcSinh}(x)$ – Santosh Linkha Dec 26 '12 at 18:29

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

• easy! nice answer +1 – Grijesh Chauhan Dec 27 '12 at 7:10

Hint: $(\sqrt{x^2+1}+x)(\sqrt{x^2+1}-x) = 1$.

• I am so surprised by your nice, simple complete answer Dan. :-) – mrs Dec 26 '12 at 18:33
• @BabakSorouh Wow, this is unexpected. Thank you very much! – Dan Shved Dec 26 '12 at 18:39
• that's why I love math – nicolas Dec 26 '12 at 19:37

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

• Interestingly, I had exactly the same answer as Adi Dani one minute before, and I get 9 upvotes while he/she gets 76... – Julien Feb 16 '13 at 20:57
• Thanks julien for your kind words. +100 – mrs Feb 19 '13 at 15:23

Another thing to add is that the Taylor series (of odd functions, if it exists) has only odd powers

$$x-{\frac {1}{6}}{x}^{3}+{\frac {3}{40}}{x}^{5}+\dots.$$

• Odd answer for the odd function. ;-) – mrs Dec 26 '12 at 18:40
• @BabakSorouh: Offcourse, we are taking about odd functions. – Mhenni Benghorbal Dec 26 '12 at 18:51

To add another hint to Dan's answer, consider that $-\log a = \log \frac1a$ and then simplify the radical out of the denominator for your function.

This function is another command for showing $f(x)=\text{arcsinh}(x)$ being increasing continous in $[0,\infty]$. We know that $f(x)=\sinh(x)$ is an odd one-one function. $$f(x)=\text{arcsinh(x)}\to \sinh(f(x))=x$$ so if $x\to -x$ then $$\sinh(f(-x))=-x\longrightarrow -\sinh(f(-x))=x\longrightarrow\sinh(-f(-x))=x=\sinh(f(x))$$ so $f(-x)=-f(x)$. This means $f(x)$ is an odd function. There is another different approach for this. See this link http://ddmf.msr-inria.inria.fr/1.8/ddmf

• +1 for your nice alternative approach, and for your kind words. – Julien Feb 19 '13 at 15:31
• I like! Helpful, too! – amWhy Mar 2 '13 at 0:49
• The link is not valid – miracle173 Feb 3 '17 at 9:57

Note by differentiating that $$f(x)=\int_0^x \frac{dt}{\sqrt{t^2+1}}.$$ The result now follows from the fact that $\dfrac{1}{\sqrt{t^2+1}}$ is even.

• It is also crucial that the integral start at $0$ and not someplace else. – whuber Dec 26 '12 at 20:53