
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

(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 $\displaystyle e^1=e=2.71...$

for (c) can't you follow the following method
$\displaystyle \frac{e^\pi}{e^\pi+1} = \frac{e^\pi+11}{e^\pi+1}$
$\displaystyle =1\frac{1}{e^\pi+1}$
$\displaystyle e^0<e^\pi<e^1$
$\displaystyle 2<e^\pi+1<e+1$
$\displaystyle 0.5>\frac{1}{e^\pi+1}>\frac{1}{e+1}$
$\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?