# What is the domain and range of the function # f(x)=5/x#?

##### 2 Answers

The domain is

The range is

#### Explanation:

In general, we start with the real numbers and then exclude numbers for various reasons (can't divide by zero and taking even roots of negative numbers being the main culprits).

In this case we cannot have the denominator be zero, so we know that

A better notation is

For the range, we use the fact that this is a transformation of a well known graph. Since there are no solutions to

**Domain** :

**Range** :

Refer to the graph attached to examine the rational function and the curve's asymptotic behavior.

#### Explanation:

A **Rational Function** is a function of the form

**The Domain :**

When dealing with the **Domain** of a Rational Function, we need to locate any points of **discontinuity** .

As these are the points where the function is not defined, we simply set

In our problem, at **discontinuity**. The curve will exhibit asymptotic behavior on either side of it.

Hence, our **Domain** :

Using **interval notation** :

We can also write our **Domain** :

That is to say the Domain includes all Real Numbers except x = 0.

Our function will *continuously approach* our **asymptote** but never quite reach that.

**The Range :**

To find the Range, let us make **x** as the subject of our function.

We will start with

Multiply both sides by **x** to get

Like we did for the **domain** , we will find out for what value(s) of **y** does the function is undefined.

We see that it is

Hence, our **Range** :

Please refer to the graph attached for a visual representation of our rational function and it's asymptotic behavior.