# Domain and Range of a Function

Want some basic practice with functions first?

The *domain* of a function
is the set of all its allowable inputs.
The *domain convention*
states that if the domain of a function
is not specified,
then it is assumed to be the
set of all real numbers for which the
function is defined.

The *range* of a function
is the set of all its outputs,
as the inputs vary through
the entire domain.

The domain of a function $\,f\,$ is denoted by $\,\text{dom}(f)\,.$

The range of a function $\,f\,$ is denoted by $\,\text{ran}(f)\,.$

Since the domain and range are
*sets*, correct set notation must be
used when reporting them.
It may be helpful to review
interval and list notation.
Remember that the symbol
$\,\mathbb{R}\,$
denotes the set of real numbers.

The domain of a function is usually quite easily determined from the formula for the function. Numbers that cause division by zero must be excluded from the domain. Anything inside an even root (square root, fourth root, etc.) must be greater than or equal to zero.

The range of a function is usually more difficult to determine from a formula. Often, it is much easier to get the range from a graph of the function (which is the topic of a future section). In this exercise, you are only asked to find the range for very simple functions.

## Examples

Using interval notation, and writing a complete mathematical sentence to report the answer: $$\cssId{s33}{\text{dom}(f)} \cssId{s34}{= [-2,\infty)}$$

- $\,f(0) = 3\,$
- $\,f(-2.79) = 3\,$
- $\,f(\pi) = 3\,$

All real numbers can be inputs. Thus, using interval notation:

$$\cssId{s46}{\text{dom}(f)} \cssId{s47}{= (-\infty,\infty)}$$Alternatively, you could write:

$$\cssId{s49}{\text{dom}(f)} \cssId{s50}{= \mathbb{R}}$$*any*nonnegative number as an output, just by taking its square as an input!

For example, suppose you want to get the output $\,1.528\,$ from the function $\,g\,.$ Just use $\,1.528^2\,$ as the input:

$$\cssId{s67}{g(1.528^2)} \cssId{s68}{= \sqrt{1.528^2}} \cssId{s69}{= 1.528}$$Using interval notation: $$\cssId{s71}{\text{ran}(g)} \cssId{s72}{= [0,\infty)}$$

So, the domain is the set
of all real numbers
*except* $\,-\frac53\,.$
That is, we want all real numbers
less than $\,-\frac53\,,$
put together with all real
numbers greater than $\,-\frac53\,.$

The ‘union’ symbol, $\,\cup\,,$ is used to ‘put sets together’. Thus:

$$\cssId{s86}{\text{dom}(f)} \cssId{s87}{= (-\infty,-\frac53)} \cssId{s88}{\cup} \cssId{s89}{(-\frac53,\infty)}$$This is the last exercise in Algebra I. (If you went through the entire course, congratulations are definitely in order!)

From here, you might want to move on to Geometry or Algebra II.