Question

How do you find the domain and range and intercepts for `H(x) = [(x - 1) (x + 2) (x - 3)]/(x(x - 4)^2)`?

Answer 1

The domain is all Real numbers with `x!=0,4`.
The intercepts are at `x=-2, 1, 3`.
The range is all Real numbers.

Explanation:

First, the domain: This is where your function (y-values) can exist. The idea here is that you can never have a `0` in the denominator of a fraction. (You can't divide anything into zero parts.) Set your denominator equal to `0` to find the vertical asymptotes, which are the places your function can't exist:

`x(x-4)^2=0` There are two terms here that can make this expression equal `0`. Either `x=0` or `(x-4)^2=0`.
So `x=0` is one of the "holes" in your domain.

Solve for the other expression by taking the square root of both sides: `(x-4)^2=0=>sqrt((x-4)^2)=sqrt0`
`x-4=0`
Add `4` to both sides, and the other place you need to exclude from the domain is at `x=4`.

As far as the intercepts, think about what those are. We often call intercepts "roots" or "zeros" of a function. They are where `y=0`. And you know that a fraction `=0` when the numerator `=0`.
Set the numerator `=0` to find your intercepts:
`(x-1)(x+2)(x-3)=0`
In this case, if any of the three terms in parentheses `=0`, the whole numerator`=0`.
If `x-1=0`, you can solve for `x` by adding `1` to both sides and get `x=1`.
Doing the same for the other two terms gives you `x=-2` & `x=3`.

Hope that answers your question!

Rational functions (these functions with the expressions in fraction form) are one of only a couple kinds of "basic" functions that don't have an infinite domain.
(Cool!)