Question

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

Answer 1

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)`