Understanding Functions and Relations

Math 100 Lesson 81: Introduction to Functions

Key Concepts

Relations

A relation is a set of ordered pairs.
Set notation uses braces (e.g., {}) to denote a relation.
Example of a relation: {(-2, 3), (1, 5), (0, 3), (6, 11)}
Domain: The set of all x-values in a relation (e.g., {-2, 0, 1, 6}).
Range: The set of all y-values in a relation, listed without repetition (e.g., {3, 5, 11}).

Functions

A function is a type of relation where no x-values repeat.
Example of a function: If x-values are {-2, 1, 0, 6}, they do not repeat, so it represents a function.
If x-values repeat, e.g., {1, -1, 1, 5}, it is not a function.

Variables

Function Notation

Denoted as f(x) which reads as "f of x."
f(x) represents the value of the function at x.
Notation could be f, g, h, etc.
To find values, substitute the given number or expression for x in the function.

Example Problems

Problem 1: Find the value of the function at 6.
- Function: f(x) = 4x + 5
- Calculate: f(6) = 4 * 6 + 5 = 29
Problem 2: Find the function value at -5.
- Function: g(x) = 3x^2 - 10
- Calculation: g(-5) = 3 * (-5)^2 - 10 = 65
Problem 3: Find the function value at a + h.
- Function: h(x) = 6x + 9
- Calculation: h(a + h) = 6(a + h) + 9 = 6a + 6h + 9

Tables and Functions

Determine if a table represents a function by checking that x-values do not repeat.
Example Table: X-values: {-2, -1, 0, 1, 2} (a function because no repeats)
Domain: X-values listed in the table.
Range: Y-values listed in the table.
Question 1: Does the table define a function?
- Evaluate by checking for unique x-values.
Question 2: Find domain and range from the table.
- Domain: {-2, -1, 0, 1, 2}
- Range: {0, 1, 3, 4, 5}
Question 3: Find the function value at a specific x (e.g., x = -1).
- Locate x in the table and find the corresponding y.
Question 4: Find x such that f(x) = 4.
- Locate 4 in the y-values and find the corresponding x-value.

Conclusion