3.6     Domain & Range            Homework: Odds 1-9, 13-41

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DSPM 0800 Class Notes

The domain of f is the set of all inputs (x-values) and the range is the set of all outputs (y-values).
· Find the domain and range from a set of ordered pairs.
Example:

The domain of f is:

The range of f is:

· Find the domain and range graphically.
Viewing a graph, the domain is the horizontal span of the graph and the range is the vertical span of the graph.
Example:
The grapher does not show this, but there are arrows on the ends of the graph. These indicate that the graph extends indefinitely left and right, as well as upward.

Since the graph continues to extend left and right (width increases indefinitely), the domain, D, is the set of all real numbers:
We see that the smallest y-value occurs at y = -3 which occurs when x = 4. The range, R, is:


Example: The following graph DOES NOT extend indefinitely, the endpoints are visible on the graph.

The domain is the horizontal span of the graph, which is from x = -2 to x = 2. The domain, D, is:

The range is the vertical span of the graph, which is from the lowest y-value at y = 0 to the largest y-value at y = 4. The range, R, is:


Example: This graph is a scatterplot of unconnected points.

The domain is the x-values of the points. The domain, D, is:

The range is the y-values of the points. The range, R, is:


· Restrictions on Domains:
1. If the outputs of f are computed by a fraction, the denominator of the fraction can never equal to zero because division by zero is undefined. Inputs that will make the denominator equal to zero are not in the domain of f. (See Interactive Discovery on page 194).

2. If the outputs of f are computed by a square root, the radicand cannot be a negative number because the square root of a negative number is not a real number. Inputs that will make the radicand negative are not in the domain of f. (See Interactive Discovery on page 195).

3. If f is used as a model for an application, the problem situation may restrict the domain.

Example: Tom is paid an hourly wage of $8. The function used to calculate the weekly wage Tom earns after working x hours per week is
           
Without knowing the problem situation, the domain would be the set of all real numbers because there are no fractions or square roots involved in the computations.
However, knowing the problem situation that x is the number of hours per week that Tom works, we know that it would not make sense for x to be a negative number. Also, after 40 hours, Tom might get overtime pay calculated at a higher rate. Thus, the domain of the function might be:


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