What is the domain and range of #y = (x^2 + 4x + 4)/( x^2 - x - 6)#?

1 Answer
Jul 4, 2017

Answer:

See below.

Explanation:

Before we do anything, let's see if we can simplify the function by factoring the numerator and denominator.

#((x+2)(x+2))/((x+2)(x-3))#

You can see that one of the #x+2# terms cancel:

#(x+2)/(x-3)#

The domain of a function is all of the #x#values (horizontal axis) that will give you a valid y-value (vertical axis) output.

Since the function given is a fraction, dividing by #0# will not yield a valid #y# value. To find the domain, let's set the denominator equal to zero and solve for #x#. The value(s) found will be excluded from the range of the function.

#x-3=0#

#x=3#

So, the domain is all real numbers EXCEPT #3#. In set notation, the domain would be written as follows:

#(-oo,3)uu(3,oo)#

The range of a function is all of the #y#-values that it can take on. Let's graph the function and see what the range is.

graph{(x+2)/(x-3) [-10, 10, -5, 5]}

We can see that as #x# approaches #3#, #y# approaches #oo#.
We can also see that as #x# approaches #oo#, #y# approaches #1#.

In set notation, the range would be written as follows:

#(-oo,1)uu(1,oo)#