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

1 Answer
Feb 22, 2018

Answer:

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!)