# Domain and Range of a Function

## Definitions of Domain and Range

### Domain

The **domain** of a
function is the complete set of possible values
of the independent variable.

In plain English, this definition means:

The domain is the set of all possible

x-values which will make the function "work", and will output realy-values.

When finding the **domain**, remember:

- The denominator (bottom) of a fraction
**cannot be zero** - The number under a square root sign
**must be positive**in this section

### Example 1a

Here is the graph of `y = sqrt(x+4)`:

The domain of this function is `x ≥ −4`, since *x* cannot be less than ` −4`. To see why, try out some numbers less than `−4` (like ` −5` or ` −10`) and some more than `−4` (like ` −2` or `8`) in your calculator. The only ones that "work" and give us an answer are the ones greater than or equal to ` −4`. This will make the number under the square root positive.

**Notes:**

- The enclosed (colored-in) circle on the point `(-4, 0)`. This indicates that the domain "starts" at this point.
- We saw how to draw similar graphs in section 4, Graph of a Function. For a more advanced discussion, see also How to draw y^2 = x − 2.

## How to find the domain

In general, we determine the **domain**of each function by looking for those values of the independent variable (usually *x*) which we are **allowed** to use. (Usually we have to avoid 0 on the bottom of a fraction, or negative values under the square root sign).

### Range

The **range** of
a function is the complete set of all possible
**resulting values** of the dependent variable (*y, *usually), after we have substituted the domain.

In plain English, the definition means:

The range is the resulting

y-values we get after substituting all the possiblex-values.

## How to find the range

- The
**range**of a function is the spread of possible*y*-values (minimum*y*-value to maximum*y*-value) - Substitute different
*x*-values into the expression for*y*to see what is happening. (Ask yourself: Is*y*always positive? Always negative? Or maybe not equal to certain values?) - Make sure you look for
**minimum**and**maximum**values of*y*. **Draw**a**sketch!**In math, it's very true that a picture is worth a thousand words.

**Example 1b**

Let's return to the example above, `y = sqrt(x + 4)`.

We notice the curve is either on or above the horizontal axis. No matter what value of *x* we try, we will always get a zero or positive value of *y*. We say the **range** in this case is *y* ≥ 0.

The curve goes on forever vertically, beyond what is shown on the graph, so the range is all non-negative values of `y`.

### Example 2

The graph of the curve *y* = sin *x * shows the **range** to be betweeen −1 and 1.

The **domain** of *y* = sin *x* is "all values of *x*", since there are no restrictions on the values for *x*. (Put any number into the "sin" function in your calculator. Any number should work, and will give you a final answer between −1 and 1.)

From the calculator experiment, and from observing the curve, we can see the **range** is *y* betweeen −1 and 1. We could write this as −1 ≤ *y* ≤ 1.

**Where did this graph come from? **We learn about sin and cos graphs later in Graphs of sin *x* and cos *x*

**Note 1: **Because we are assuming that only real numbers are to be used for the *x*-values, numbers that lead to **division by zero** or to **imaginary numbers** (which arise from finding the square root of a negative number) are not included. The Complex Numbers chapter explains more about imaginary numbers, but we do not include such numbers in this chapter.

**Note 2: **When doing square root examples, many people ask, "Don't we get 2 answers, one positive and one negative when we find a square root?" A square root has at most one value, not two. See this discussion: Square Root 16 - how many answers?

**Note 3: **We are talking about the domain and range of **functions**, which have **at most** one *y*-value for each *x*-value, not **relations** (which can have more than one.).

## Summary

In general, we determine the **domain** by
looking for those values of the independent variable (usually *x*) which we are **allowed** to use. (We have to avoid 0 on the bottom of a fraction, or negative values under the square root sign).

The **range** is found by finding the resulting *y*-values after we have substituted in the possible *x*-values.

### Exercise 1

Find the domain and range for each of the following.

(a) `f(x) = x^2+ 2`.

### Need Graph Paper?

(b) `f(t)=1/(t+2)`

(c) `g(s)=sqrt(3-s)`

(d) `f(x) = x^2+ 4` for `x > 2`

### Exercise 2

### More Domain and Range Examples

In case you missed it earlier, you can see more examples of domain and range in the section Inverse Trigonometric Functions.

We fire a ball up in the air and find the
height *h*, in metres, as a function of time
*t*, in seconds, is given by

h= 20t− 4.9t^{2}

Find the domain and range for the function
*h*(*t*).

## Functions defined by coordinates

Sometimes we don't have continuous functions. What do we do in this case? Let's look at an example.

### Exercise 3

Find the domain and range of the function defined by the coordinates:

`{(−4, 1), (−2, 2.5), (2, −1), (3, 2)}`

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