Inverse Functions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 2 November 2009

Inverse Functions

[???] SHOULD WE SAY SOMETHING ABOUT PRINCIPAL DOMAINS HERE?

$\sqrt{x}$ is the function that undoes $x^2$. This means that $$\sqrt{x^2}=x\text{and}{\left(\sqrt{x}\right)}^2=x\text{.}$$ [???] FIRST EQN ONLY TRUE FOR x POSITIVE

$\ln x$ is the function that undoes $e^{\mathrm{x}}$. This means that $$\ln \left(e^x\right)=x\text{and}{e}^{\ln x}=x\text{.}$$ [???] SECOND EQN ONLY TRUE FOR x POSITIVE (REAL DEFN OF ln)

${\sin }^{-1}x$ is the function that undoes $\sin x$. This means that $${\sin }^{-1}\left(\sin x\right)=x\text{and}\sin \left({\sin }^{-1}\right)=x\text{.}$$ [???] SECOND EQN ONLY TRUE FOR x IN [-pi/2, pi/2]

${\cos }^{-1}x$ is the function that undoes $\cos x$. This means that $${\cos }^{-1}\left(\cos x\right)=x\text{and}\cos \left({\cos }^{-1}\right)=x\text{.}$$

${\tan }^{-1}x$ is the function that undoes $\tan x$. This means that $${\tan }^{-1}\left(\tan x\right)=x\text{and}\tan \left({\tan }^{-1}\right)=x\text{.}$$

${\cot }^{-1}x$ is the function that undoes $\cot x$. This means that $${\cot }^{-1}\left(\cot x\right)=x\text{and}\cot \left({\cot }^{-1}\right)=x\text{.}$$

${\sec }^{-1}x$ is the function that undoes $\sec x$. This means that $${\sec }^{-1}\left(\sec x\right)=x\text{and}\sec \left({\sec }^{-1}\right)=x\text{.}$$

${\csc }^{-1}x$ is the function that undoes $\csc x$. This means that $${\csc }^{-1}\left(\csc x\right)=x\text{and}\csc \left({\csc }^{-1}\right)=x\text{.}$$

${\log }_{a}x$ is the function that undoes $a^x$. This means that $${\log }_a\left(a^x\right)=x\text{and}{a}^{{\log }_{a}x}=x\text{.}$$

WARNING: ${\sin }^{-1}x$ is VERY DIFFERENT from ${\left(\sin x\right)}^{-1}$. For example, $${\sin }^{-1}0={\sin }^{-1}\left(\sin 0\right)=0\text{,}\text{BUT}{\left(\sin x\right)}^{-1}=\frac{1}{\sin 0}=\frac{1}{0}$$ which is undefined.

Example. Explain why $\ln 1=0$. $$\ln 1=\ln \left(e^0\right)=0\text{.}$$

Example. Explain why $\ln \left(ab\right)=\ln a+\ln b$. $$\ln \left(ab\right)=\ln \left(e^{\ln a}·e^{\ln b}\right)=\ln \left(e^{\ln a+\ln b}\right)=\ln a+\ln b\text{.}$$

Example. Explain why ${\displaystyle \ln \left(\frac{1}{a}\right)=-\ln a}$. $$\ln \left(\frac{1}{a}\right)=\ln \left(\frac{1}{e^{\ln a}}\right)=\ln \left(e^{-\ln a}\right)=-\ln a\text{.}$$

Example. Explain why $\ln \left(a^b\right)=b\ln a$. $$\ln \left(a^b\right)=\ln \left({\left(e^{\ln a}\right)}^b\right)=\ln \left(e^{b\ln a}\right)=b\ln a\text{.}$$

Thus

$e^0=1$	turns into	$\ln 1=0$,
$e^x{e}^y=e^{x+y}$	turns into	$\ln ab=\ln a+\ln b$,
${\displaystyle e^{-x}=\frac{1}{e^x}}$	turns into	${\displaystyle \ln \frac{1}{a}=-\ln a}$,
${\left(e^x\right)}^y=e^{yx}$	turns into	$\ln \left(a^b\right)=b\ln a$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)