# How to find domain, asymptotes, holes, intercepts for #f(x) = x / (3x(x-1))#?

##### 1 Answer

Domain:

Asymptote:

Hole:

X-intercept: None

Y-intercept: None

#### Explanation:

Obvious temptation is to reduce the fraction by

But, prior to do that, we have to make sure that the function does not change its domain by this transformation. That means, we have to specify that the new function has exactly the same domain as the original one.

The original function has two points, *domain* as

Another representation of the same domain would be

or

or

Now, when we properly restricted the domain, we can reduce the fraction by

This is a familiar hyperbola shifted to the right by *hole*.

Our hyperbola has an *asymptote* at

It does not intersect the X-axis, so there is no X-intercept.

To find Y-intercept, we have to find the value of this function at