# Range of a function

The range of a function is the set of all possible values it can produce.
For example, consider the function No matter what value we give to x, the function is always positive:

• If x is 2, then the function returns x squared or 4.
• If x is negative 2, then it still produces 4 since -2 times -2 is positive 4.

So the range of the function is

"all real numbers greater than or equal to zero".

## Notation

Ranges can be written out in words as above, but to be more mathematically precise they are also written using either inequalities, or in interval notation:

• As an inequality, we would write Which is read as "the function f(x) has a value which is always greater than or equal to zero".
For more on inequalities see Inequalities.

• In so-called interval notation, the same function has a range of This describes the range of values from 0 to positive infinity. The square brackets means the range includes zero and infinity themselves.

## As a graph

If we plot the function above on a graph we see that for any value of x, the graph is either zero or positive - above the x axis.

While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at   patreon.com/mathopenref