Question

If f(x)=(√(2-x)) and g(x)= (x-1), then find the domain of h(x)=f(x)/g(x)

Answer 1

The function is $\displaystyle{ \frac{ \sqrt{2-x}}{x-1}$. 

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, $2-x \geq 0$, is solved as follows:

adding -2 to both sides we have $-x \geq -2$.


Multiplying by -1 (and not forgetting to swap the sign), we have $x \leq 2$.


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]