# What is the domain and range of  f(x) = x / (3x(x-1))?

Jan 13, 2018

Domain f(x): $\left\{x \epsilon \mathbb{R} | x \ne 0 , 1\right\}$

#### Explanation:

In order to determine the domain, we need to see which part of the function restricts the domain. In a fraction, it is the denominator. In a square root function, it is what's inside the square root.

Hence, in our case, it is $3 x \left(x - 1\right)$.

In a fraction, the denominator can never be equal to 0 (which is why the denominator is the restricting part of the function).

So, we set:
$3 x \left(x - 1\right) \ne 0$

The above means that:
$3 x \ne 0$ AND $\left(x - 1\right) \ne 0$

Which gives us:
$x \ne 0$ AND $x \ne 1$

Thus, the domain of the function is all real numbers, EXCEPT $x = 0$ and $x = 1$.

In order words, domain f(x): $\left\{x \epsilon \mathbb{R} | x \ne 0 , 1\right\}$