Question

Determine the maximal domain and corresponding range of the function with the rule: f(x)=log base e (3+3cos((pi/2)x))

Answer 1

The domain of \(\ln f(x)\) is \(f(x)>0\), so you must determine for which values of \(x\) you have
\[3+3\cos\frac{\pi}{2}x>0~~\iff~~\cos\frac{\pi}{2}x>-1\]
Recall that \(-1\le\cos ax\le1\). This means you need to find the values of \(x\) that make \(\cos \dfrac{\pi}{2}x=-1\). This cosine function is always greater than -1 except for when it is exactly -1.
\[\cos\frac{\pi}{2}x=-1~~\Rightarrow~~\frac{\pi}{2}x=\pi+2n\pi~~\Rightarrow~~x=2+4n\]
for integer \(n\).

Since \(-1\le\cos ax\le1\), you have
\[-1\le\cos\dfrac{\pi}{2}x\le1\\
-3\le3\cos\frac{\pi}{2}x\le3\\
0\le3+3\cos\frac{\pi}{2}x\le6\\
-\infty<\ln\left(3+3\cos\frac{\pi}{2}x\right)\le\ln6\]
Here's a plot of the function with its maximum:
http://www.wolframalpha.com/input/?i=Log%5B3%2B3*Cos%5Bpi%2F2*x%5D%5D%2C+Log%5B6%5D