Domain

The domain is the set of all possible input values (usually the x-values) that produce a valid output. It represents the "starting numbers" we can use.

Range

The range is the set of all possible output values (usually the y-values or f(x) values) that result from using the relation or function. It represents the "ending numbers" we can get.

From Ordered Pairs and Arrow Diagrams

When a relation is given as a set of ordered pairs or shown in an arrow diagram, we find the domain and range by listing the values that appear.

Example 1:

Consider the relation R = {(1, a), (1, b), (3, a), (3, c)}

• Domain: Collect all first coordinates: {1, 3}
• Range: Collect all second coordinates: {a, b, c}

Example 2:

Let P = {1, 3, 4, 10} and Q = {0, 4, 8}. The relation R: "is less than" connects elements from P to Q.

From the arrow diagram:

• Domain: {1, 3, 4} — these elements in P have a number in Q that is greater than them
• Range: {4, 8} — these are the outputs that appear

From Algebraic Functions with Restricted Domains

When a function is given with a rule like y = f(x) and restrictions on x, we find the domain from the restrictions and the range by calculating the output values.

Example 3:

Let R = {(x, y) : y = x + 1 and -2 < x ≤ 8}, where x and y are integers.

Finding the domain: The domain is restricted by -2 < x ≤ 8, so:

• Domain = {-1, 0, 1, 2, 3, 4, 5, 6, 7, 8}

Finding the range: Since y = x + 1, we add 1 to each domain value:

• Range = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

From Graphs

When a relation or function is shown as a graph, we read the domain from the x-axis and the range from the y-axis.

Key points:

• The domain includes all x-values covered by the graph
• The range includes all y-values covered by the graph

Example 4:

A function is graphed and covers x-values from -3 to 5, and y-values from -2 to 4.

• Domain: -3 ≤ x ≤ 5, or [-3, 5]
• Range: -2 ≤ y ≤ 4, or [-2, 4]

To check if a graph represents a function, use the vertical line test: if a vertical line crosses the graph at more than one point, it is not a function.

From Tables

When data is presented in a table, the domain consists of all input values (first row or column), and the range consists of all output values (second row or column).

Example 5:

• Domain: {-2, -1, 0, 1, 2}
• Range: {0, 1, 4}

1. Identify what represents inputs (x) and outputs (y) in the given representation
2. Collect all possible input values → this is the domain
3. Collect all possible output values → this is the range
4. Express the domain and range using set notation or interval notation

In Tanzania, understanding domain and range is useful when analyzing business data. For example, a shop owner tracking daily sales might record the number of customers (domain) and the corresponding revenue in Tanzanian shillings (range). By examining this relation, the owner can determine the minimum and maximum revenue expected for a given number of customers, helping with planning inventory and setting sales targets.