Recall that we defined h to be the height of a ball t seconds after it has been thrown into the air. Although h is a function of t, the value of h is not defined for every possible value of t.

For example, it makes no sense to talk about the height of the ball when t = -2 or t = 4. So, although h is a function of t, this function is only defined for certain values of t.

Also, notice that h, the output value of the function, can only attain values between 0 and 30.

The set of input values for which the function is defined is called the domain of the function.

Now, consider the function h(x), whose graph is shown. We want to find the domain and range of this function.

By looking at the graph, you can see that the domain of this function is all real numbers. The set of real numbers can be represented as the open interval from negative infinity to infinity.

The parentheses next to the infinity symbols mean that infinity and negative infinity are not included in the domain. This is because these values are not actually numbers.

Now, let's find the range of h(x). The range of this function is all real numbers greater than zero.

Again, the parenthesis next to zero means that zero is not included in the range. Similarly, the parenthesis next to the infinity symbol means that infinity is not included in the range.

When using interval notation, we use the bracket symbol to indicate that an endpoint of the interval is included and a parenthesis to indicate that an endpoint is not included.

The graph of function f(x) is displayed to the right. Use the graph to help you find the domain and the range of f(x). Click "Submit" when finished.

You have seen how to determine the domain and range of a function using its graph. Now, let's see how you can find the domain and range of a function by studying its equation algebraically.

Consider the function . Recall that the domain of a function is all real numbers except those which do not yield an output value.

Division by zero is undefined, so the expression is defined for any real number except zero. Therefore, the domain of f(x) is all real numbers except zero.

The range of a function is all the real numbers that it can return as output values.

Dividing 1 by any real number is never zero. So, it is not possible for f(x) to equal zero. All real numbers except zero are possible output values because all nonzero real numbers have reciprocals. So, the range of f(x) is all real numbers except zero.

The graph of f(x) confirms this domain and range.

Find the domain of the function by examining its equation. Click "Submit" when finished.

Find the range of the function by examining its equation. Click "Submit" when finished.

The corresponding set of output values is called the range of the function.

So the domain of the function in our example is the set of all real numbers between 0 and 2.619. This set can also be described using interval notation, as shown.

The bracket next to 0 means that zero is included in the domain. Similarly, the bracket next to 2.619 means that 2.619 is included in the domain.

The range of the function is the set of all real numbers between 0 and 30. Again, the brackets mean that 0 and 30 are included in the range.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education