The function is [tex]\displaystyle{ \frac{ \sqrt{2-x}}{x-1} [/tex].
The domain can only contain those values of x for which the denominator is not zero, that is all real numbers except 1,
AND for which 2-x i greater or equal to 0 (because if 2-x is negative, the square root of it cannot be calculated).
The last inequation, [tex]2-x \geq 0[/tex], is solved as follows:
adding -2 to both sides we have [tex]-x \geq -2[/tex].
Multiplying by -1 (and not forgetting to swap the sign), we have [tex]x \leq 2[/tex].
So x must be smaller or equal than 2, but also x cannot be 1, from the first condition. This means that the answer is ( - ∞ , 1 ) U ( 1 , 2].
Answer: ( - ∞ , 1 ) U ( 1 , 2]
