# Ranges of a function

• Nov 22nd 2010, 06:03 PM
sirellwood
Ranges of a function
If $\displaystyle \pi$ $\displaystyle \in$ (0,1), state the ranges of

(a) log($\displaystyle \pi$)

(b) log($\displaystyle \pi$/(1−$\displaystyle \pi$))

(c) exp($\displaystyle \pi$)/(1+exp($\displaystyle \pi$))

Anyone know how to find these?

Thanks
• Nov 22nd 2010, 06:06 PM
dwsmith
(a) $\displaystyle (-\infty, 0]$ It may help to look at a graph

(b) use properties of logs to break up this log

(c) a graph may help here $\displaystyle e^0=1$ and $\displaystyle e^1=e=2.71...$
• Nov 23rd 2010, 01:35 AM
for (c) can't you follow the following method

$\displaystyle \frac{e^\pi}{e^\pi+1} = \frac{e^\pi+1-1}{e^\pi+1}$

$\displaystyle =1-\frac{1}{e^\pi+1}$

$\displaystyle e^0<e^\pi<e^1$

$\displaystyle 2<e^\pi+1<e+1$

$\displaystyle 0.5>\frac{1}{e^\pi+1}>\frac{1}{e+1}$

$\displaystyle 0.5<1-\frac{1}{e^\pi+1}<1-\frac{1}{e+1}$

$\displaystyle R(f)=(0.5,(1-\frac{1}{e+1}))$

(if $\displaystyle f$ is the function)
am I correct?