Question

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

Answer is:              ( - ∞ , 1 ) U ( 1 , 2]

Answer 1

The function is
`\displaystyle{ ( √(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]