1. Section 5.2
Functions

2. Relation, Domain, and Range
Definition
A relation is a set of ordered pairs. The domain of a relation is the set of all values of the independent variable, and the range of the relation is the set of all values of the dependent variable.
We can think of a relation as a machine in which values of x are “inputs” and values of y are “outputs.” In general, each member of the domain is an input, and each member of the range is an output.

3. A relation described by a table.
Relation, Domain, and Range
A relation described by a table.
A relation described by a graph.

4. Think of a relation as an input-ouput machine.
Relation, Domain, and Range
Think of a relation as an input-ouput machine.

5. Function
Definition
A function is a relation in which each input leads to exactly one output.

6. Deciding whether an Equation Describes a Function
Example 1
Is the relation y = x + 2 a function? Find the domain and range of the relation.
Solution
Let’s consider some input-output pairs in the table on the next slide.

7. Each input leads to just one output-namely, the input
Deciding whether an Equation Describes a Function
Solution continued
Each input leads to just one output-namely, the input
increased by 2-so the relation y = x + 2 is a function.

8. Deciding whether an Equation Describes a Function
Solution continued
The domain of the relation y = x + 2 is the set of all real numbers, since we can add 2 to any real number. The range of y = x + 2 is also the set of real numbers, since any real number is the output of the number that is 2 units less than it.

9. Is the relation y = ±x a function?
Deciding whether an Equation Describes a Function
Example 2
Is the relation y = ±x a function?
Solution
If x = 1, then y = ±1. So, the input x = 1 leads to two outputs: y = −1 and y = 1. Therefore, the relation y = ±x is not a function.

10. Is the relation y2 = x a function?
Deciding whether an Equation Describes a Function
Example 3
Is the relation y2 = x a function?
Solution
Let’s consider the input x = 4.
The input x = 4 leads to two outputs: y = −2 and y = 2. So, the relation y2 = x is not a function.

11. Is the relation described by the table a function?
Deciding whether a Table Describes a Function
Example 4
Is the relation described by the table a function?
Solution
The input x = 1 leads to two outputs: y = 3 and y = 5. So, the relation is not a function.

12. Vertical-Line Test
A relation is a function if and only if every vertical line intersects the graph of the relation at no more than one point. We call this requirement the vertical-line test.

13. Determine whether the graph represents a function.
Deciding whether a Graph Describes a Function
Example 6
Determine whether the graph represents a function.

14. Deciding whether a Graph Describes a Function
Solution
1. Since the vertical line (red) sketched intersects the circle more than once, the relation is not a function.

15. Deciding whether a Graph Describes a Function
Solution
2. Each vertical line sketched intersects the curve at one point. In fact, any vertical line would intersect this curve at just one point. So, the relation is a function.

16. Finding the Domain and Range
Example 9
Use the graph of the function to determine the function’s domain and range. 1. 2.

17. Finding the Domain and Range
Solution
The domain is the set of all x-coordinates of points in the graph. Since there are no breaks in the graph, and since the leftmost point is (−4, 2) and the rightmost
point is (5,−3), the domain is −4≤ x ≤ 5. The range is the set of all y-coordinates of points in the graph. Since the lowest point is (5, −3) and the highest point is (2, 4), the range is −3 ≤ y ≤ 4.

18. Finding the Domain and Range
Solution continued
The graph extends to the left and right indefinitely without breaks, so every real number is an x-coordinate of some point in the graph. The domain is the set of all real numbers.

19. Finding the Domain and Range
Solution continued
The output −3 is the smallest number in the range, because (1, −3) is the lowest point in the graph. The graph also extends upward indefinitely without breaks, so every number larger than −3 is also in the range. The range is y ≥ −3.