Section 4.1: Introduction to Functions
A function is a relation in which each input has exactly one output. Functions are used to model relationships between quantities.
Definition:
A function \( f \) from set \( X \) to set \( Y \) assigns each element \( x \) in \( X \) exactly one element \( y \) in \( Y \), denoted \( f(x) = y \).
Example 1
Determine whether the relation \(\{(1,2),(2,3),(3,4),(4,5)\}\) is a function.
Each input (1,2,3,4) maps to exactly one output, so it is a function.
Example 2
Determine whether the relation \(\{(1,2),(2,3),(1,4),(3,5)\}\) is a function.
The input 1 maps to two outputs (2 and 4), so it is not a function.
Practice Problems
- Determine if \(\{(0,1),(1,2),(2,3),(3,4)\}\) is a function.
- Determine if \(\{(1,2),(2,3),(1,3),(4,5)\}\) is a function.
- Describe in words what it means for a relation to be a function.
- Give an example of a function with domain \(\{a,b,c\}\) and range \(\{1,2,3\}\).
- Give an example of a relation that is not a function.