What is the domain of a function that contains radicals and rational functions all together? 
In the picture, I tried to find the domain of each separate function but I do not understand if this is the right process and how to combine the resulting domains.The function is $(\frac{1-x}{x-3})^{1/2}+(cos{(πx)})^{1/2}$ .
 A: I liked this question a lot. But you should spend a little time and learn mathjax.
I’m tutoring in a couple of high-school classes that are worrying over exactly this kind of question. First, I should say that as given, it’s not a function, it’s just a mathematical expression, which may or may not make sense. When you identify all numbers for which it does make sense, then you may use this identification to define a set, and proclaim a function whose domain is this set, and whose rule is given by your expression.
Now, there’s a serious difficulty with what you have presented us, because your manuscript says $\sqrt{1-x}\big/\sqrt{x-3}$, while your typescript says $\sqrt{\frac{1-x}{x-3}}$. You may be surprised to see that these two expressions do not say the same thing! 
For your specific expression
$$
\sqrt{\frac{1-x}{x-3}}+\cos(\pi x)\,,
$$
the second term, the trigonometric, poses no problem, since it makes sense for all real $x$. It’s the radical part that makes you stop. First, everything under a radical has to be nonnegative, i.e. $\ge0$, and when you examine that fraction more closely, you see that it’s not possible to get both top and bottom simultaneously positive. But if they’re simultaneously negative, then the radicand (what’s under the square root sign) becomes positive all right. Remembering that the denominator can’t be zero, you get two inequalities $1-x\le0$ and $x-3<0$, to which you apply standard techniques to get $1\le x$ and $x<3$. They both have to be true. You can describe the interval on which the expression gives you a good function.
