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

Jul 4, 2017

See below.

#### Explanation:

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

$\frac{\left(x + 2\right) \left(x + 2\right)}{\left(x + 2\right) \left(x - 3\right)}$

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

$\frac{x + 2}{x - 3}$

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:

$\left(- \infty , 3\right) \cup \left(3 , \infty\right)$

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 $\infty$.
We can also see that as $x$ approaches $\infty$, $y$ approaches $1$.

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

$\left(- \infty , 1\right) \cup \left(1 , \infty\right)$