Question

y=sqrt{cosx}+sqrt{25-x^2}
Domain of y=?

Answer 1

I think we need the graph method for this,im confused about that.

Answer 2

Since you can only take the square root of a number \(\geq\) 0,
\(\cos x \geq 0\) and \(25 - x^2 \geq 0\)
For the latter one,
\(x^2 \leq 25\), \(-5 \leq x \leq 5\)

Since we have one root satisfied now we'll look at the other one, taking into account we only have to look on the interval [-5, 5], also note that
\(\Large -2\pi < -5 < -\frac{3\pi}{2} \mbox{ and } \frac{3\pi}{2} < 5 < 2\pi\)
cos(x) is \(\geq\) 0 on \(\Large [-{\pi \over 2}, {\pi \over 2}]\) \(\Large [\frac{3\pi}{2}, 2\pi]\) and \(\Large [-\frac{3\pi}{2}, 2\pi]\)

\(\Large -5 \leq x \leq -\frac{3\pi}{2} \mbox{ or } -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \mbox{ or } \frac{3\pi}{2} \leq x \leq 5 \)