Finding the Domain of a Function Step by Step

What is the domain of a function? In what is a function we saw that we can think of a function as a rule. This rule, given an input, gives an output.

You could also say that it is a rule that "maps" the input to the output. In calculus, we often use this notation:

y function of x

This means that "y" is a function of "x". Here, x is the input, often called the independent variable. And y is the output, and it is called the dependent variable.

So, back to the domain of a function. What is it? It is just the set of values that x can take. You can think of the domain as a bag. This bag contains all the x's you can choose as input for the function.

The domain of a function can be defined explicitly or implicitly, but it is always defined.

Example 1

As an example of a domain defined explicitly, let's say I give you the expression:

function with explicit domain

Here I tell you that x must be greater than 0. You can't choose any x. In the bag you only have positive x's.

On the other hand, if I simply tell you that:

y function of x

This function has an implicitly defined domain. I don't specify the valid values of x. So, it is implicit that the domain is the set of all real numbers.

Example 2

A more interesting example of an implicitly defined domain is the function:

function with implicit domain

At first glance you may think this is the same as the previous case. However, what would happen if x=2? We'll get

function with implicit domain

And 1/0 doesn't make sense. (if you're not completely sure why division by zero doesn't make sense, here's a simple explanation).

Because f(2) doesn't make sense, we take the 2 out of the bag, and the domain is the set of all real numbers that are not 2.

More examples...

Example 3

Let's consider the function:

function with square root

In the real numbers, the squareroot of a number is defined only for positive numbers. The squareroots of negative numbers do exist, but we won't consider them here.

So, our function is only defined when there is a positive number inside (or zero!) the square root sign. That means that x-3 must be positive:

function with square root

Adding 3 to both sides:

finding the domain

And the domain of this function is the set of all numbers equal or greater than 3. A fancy way of saying this is that the domain is:

the domain in interval notation

This is an interval that includes 3 and all numbers greater than 3.


  • The domain of a function is the set of all values the independent variable can take.
  • The domain can be specified explicitly or implicitly. When it is implicit, the domain is the set of all real numbers for which the function makes sense.

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