# How do you find the domain and range of #f(x)= 1/x + 5/(x-3)#?

##### 1 Answer

Domain:

All real numbers

In other words,

Range:

All real numbers.

In other words,

#### Explanation:

It is traditionally assumed that functions like this, unless specifically mentioned otherwise, are defined for *real numbers* as argument, having values also among *real numbers*.

**Domain** of a real function is a set of values where this function is defined.

The function

This happens only for

Therefore, the domain of this function is:

all real values except

It can be written as

Alternatively, it can be written as

One more way:

**Range** of a real function is a set of values that this function can take while its argument takes all the values from the **domain**.

To determine the range, let's try to resolve an equation

for any value

So, let's try to find all

We assume that

Multiplying the equation by

or

The quadratic equation above has a solution if its discriminant is not negative.

The discriminant of this equation is

As we see, the discriminant

An interesting exercise would be to graph this function. I suggest to add two graphs,

graph{1/x + 1/(x-3) [-10, 10, -5, 5]}