Find the domain of the following functions
(a) `f(x)=(1)/(sqrt(x-2)) " (b) " f(x)=(1)/(x^(3)-x)`
(c ) `f(x)= root(3)(x^(2)-2)`

Answer

(a) `(2,oo) " (b) " R-{-1,0,1} " (c ) "R`

Explanation

(a) `f(x)=(1)/(sqrt(x-2))` is defined if `x-2 gt 0 " or " x gt 2.`
Therefore, domain is `(2,oo)`.
(b) `f(x)=(1)/(x^(3)-x)` is not defined if `x^(3)-x=0 " or " x(x-1)(x+1)=0" or " x=-1,0,1.`
Therefore, domain is `R-{-1,0,1}.`
(c ) `f(x)=root(3)(x^(2)-2)`. We know that cube roots are defined for any real value. So, `x^(2)-2` can take any real value.
So, x can take any real value. Therefore, domain is set R.