# How do you find the domain and range of #y = (x - 3)/( x^2 - 4)#?

##### 2 Answers

Domain:

or

Range:

#### Explanation:

Domain is easy. This function is defined everywhere except those values of argument

or

Solutions are

At these points the function has vertical asymptotes.

To address the range, let's first transform the function as follows:

Next step is to graph this function as a sum of two graphs *Algebra - Graphs*.

The resulting graph would look like

graph{1/(x+2)-1/(x^2-4) [-10, 10, -5, 5]}

It's easy to see from this graph that the only segment not covered by values of this function is the one between the maximum of the right part of a graph and minimum of the middle part.

To find these values, let's use the calculus to find the arguments where our function reaches its local maximum and minimum values, that is those values of

Taking the derivative of this function results in'

Now we have solve the equation

or

or

or

In this case we have to find where the numerator equals to zero, that to solve

or

or

Solutions of this quadratic equation are

We have to use the larger value

We have to use the smaller value

Therefore, the range of our function is

Solve for

#### Explanation:

Here's an alternative method to try to find the range using elementary methods.

Multiply both sides by

Subtract

If

Otherwise, use the quadratic formula to get:

This will have Real solutions when the discriminant is non-negative, that is when:

This quadratic in

Hence