The domain of a **function** is the set of *x*-coordinates of the points in the function. The range of the function *f* is the set of y-coordinates of the points in the function. So if we have a function *f* with points (-3, -2), (-1, 3), (2, 3), and (5,4), then the domain of the function *f* is the set {-3, -1, 2, 5} and the range of *f* is the set {-2, 3, 4). Graphically, we can say that the domain is the “projection” of the points to the x-axis (see red points in the following figure).

The range of *f* is the projection of the points to the y-axis (see green points in the following figure).

**Example 1**

In the function *g*, we can see that the projection of the graph to the *x*-axis is the set of **real numbers** from -1 to 2 (see red line segment), however, small “empty circle” indicates that the domain does not include 2. Therefore, the domain of g is the set of real number

.

In interval notation, that is .

**Recall:** The interval notation where *a* is less than *b*, means from *a* to *b* including *a* and *b* and means from *a* to *b* excluding *a* and excluding *b*. The interval means from a to b excluding a and including b and means from *a* to *b* including *a* and excluding *b*.

Looking at the graph, the projection of the graph to the y-axis is from -2.7 up to 3.5 but not including 3.5 because of the empty circle (see green line segment). Therefore, the range is the set

or in interval notation.

**Example 2**

In the function *h*, the projection of the graph to the *x*-axis is from -3 to 2, except -2. Therefore, the domain of *h* is the set of real numbers from -3 to 2, excluding -2. Using the interval notation, we can them into two intervals which are from -3 to 2 and from -2 to 2 excluding -2 on both intervals.

In interval notation, we can say that the domain of h is .

**Example 3**

The projection of the graph on the x-axis is on the entire x-axis (can you visualize why?), so its domain is the set of real numbers. Its projection on the y-axis is from 0 extending upward indefinitely, so the range of f is the set of non-negative real numbers or in interval notation.

**Example 4**

The graph of the linear function is a slanting line extending to the left and right indefinitely as well as up and down indefinitely. This means that the domain of *q* is the set of real numbers and the range is also the set of real numbers.

**Example 5**

The domain of the sine function whose graph is shown below is the set of real numbers (why?). Its range is from -1 to 1.

Now, you have a conceptual understanding of the graphical representation of domain and range. In the **next post**, we are going to learn some algebraic methods on how to find the domain and range of functions.