Section 1.8: Functions & Relations
A relation is any set of ordered pairs. A function is a special type of relation where each input has exactly one output.
Identifying Functions:
- Use the vertical line test: If any vertical line crosses a graph more than once, it is not a function.
- In a table or mapping, no input value should be paired with more than one output.
Example 1
Is the relation \( \{(1,2), (2,3), (3,4), (4,5)\} \) a function?
Each input has exactly one output. Yes, it is a function.
Example 2
Is the relation \( \{(1,2), (1,3), (2,4)\} \) a function?
The input 1 is paired with two outputs (2 and 3). This violates the definition of a function.
Therefore, it is not a function.
Example 3
Does the graph of a circle represent a function?
No. A vertical line can intersect the circle in two points, so it fails the vertical line test.
Practice Problems
- Determine if \( \{(2,5), (3,7), (4,9)\} \) is a function.
- Check whether the relation \( \{(0,1), (0,-1), (2,3)\} \) is a function.
- Does the equation \( y = x^2 \) define a function?
- Does the graph of a parabola represent a function?
- Explain the vertical line test in your own words.