What is the domain of the given function with the greatest integer?

Question

48. The domain of the function

$$f(x)=\sqrt{\frac{4-x^2}{[x]+2}}$$

where $[x]$ represents the greatest integer function, is

(a) $(-\infty,-1)\cup[-1,2]$
(b) $(-\infty,-2)\cup[0,2]$
(c) $(-\infty,-2)\cup[-1,2]$
(d) None of the above

If we put $-1.5$ then we get $0$ in denominator so option a is wrong, but how can I check other options b and c?

Answer 1

First notice that the numerator $4-x^2\geq 0$ when $-2\leq x \leq 2$, and less than or equal to zero elsewhere.

You should then discuss by cases:

1. When $x\leq -2$, both the denominator and numerator are less than or equal to zero. The function is well-defined.

2. When $-2 \leq x \leq -1$, the denominator is 0, undefined.

3. When $-1\leq x \leq 2$, the denominator and numerator are both positive, so it is well-defined.

4. When $x\geq 2$, the denominator is positive. The numerator is negative. It is undefined.