Relations, Functions, Domain & Range, and Piecewise Functions: Calculus Foundations

Relations and Functions

Definition of Relations and Functions

Understanding the concepts of relations and functions is fundamental in calculus and higher mathematics. These concepts describe how elements from one set (inputs) are associated with elements from another set (outputs).

• Relation: A relation is a connection between two sets of values, typically represented as ordered pairs $(x, y)$.

• Function: A function is a special type of relation where each input (x-value) is associated with at most one output (y-value).

Identifying Functions

To determine whether a relation is a function, use the following criteria:

• Vertical Line Test: If any vertical line drawn through the graph passes through more than one point, the graph is not a function.

• Inputs and Outputs: Each input must correspond to only one output for the relation to be a function.

Examples

• Example 1: The set $ig\{(-3, 5), (0, 2), (3, 5)ig\}$ is a function because each input has only one output.

• Example 2: The set $ig\\{(2, 5), (0, 2), (2, 9)ig\}$ is not a function because the input 2 corresponds to two different outputs (5 and 9).

Function vs. Non-Function (Graphical)

• Function: Graphs that pass the vertical line test (e.g., a parabola).

• Not a Function: Graphs that fail the vertical line test (e.g., a circle).

Domain and Range

Definition

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Domain: Allowed x-values for the function.

• Range: Allowed y-values for the function.

Finding Domain and Range from a Graph

• To find the domain, project the graph onto the x-axis.

• To find the range, project the graph onto the y-axis.

Notation

• Interval Notation:

  • Square brackets [ ] mean the endpoint is included.

  • Parentheses ( ) mean the endpoint is not included.

• Set Builder Notation:

  • Domain: $\{x \mid -4 \leq x < 5\}$

  • Range: $\{y \mid -1 \leq y < 2\}$

Example

• Example: For a graph that extends from $x = -4$ to $x = 5$ and $y = -1$ to $y = 2$, the domain is $[-4, 5)$ and the range is $[-1, 2)$.

Multiple Intervals

• When a graph has multiple intervals, use the union symbol $\cup$ to combine them.

Piecewise Functions

Definition

A piecewise function is a function defined by different equations over different intervals of the domain.

• Each 'piece' applies to a specific interval of x-values.

• If the values at the boundaries do not match, the function has a jump (discontinuity) at those points.

Notation and Evaluation

• Piecewise functions are written as: $f(x) = \begin{cases} 4 & \text{if } x < 3 \\ x^2 - 3 & \text{if } 3 \leq x < 8 \\ x^3 & \text{if } x \geq 8 \end{cases}$

• To evaluate $f(a)$, determine which interval $a$ falls into and use the corresponding formula.

Example

• Example: For $f(x) = \begin{cases} -4 & \text{if } x < -2 \\ x + 1 & \text{if } -2 \leq x < 1 \\ x^2 & \text{if } x \geq 1 \end{cases}$, to find $f(0)$, use the second piece: $f(0) = 0 + 1 = 1$.

Practice Problems

• Evaluate $f(-1)$ for $f(x) = \begin{cases} 4 & x < 3 \\ x^2 - 3 & 3 \leq x < 8 \\ x^3 & x \geq 8 \end{cases}$

• Evaluate $f(5)$ for $f(x) = \begin{cases} 2x + 4 & x < 0 \\ x^2 & 0 \leq x < 2 \\ x^2 + 5 & x \geq 2 \end{cases}$

Graphing Piecewise Functions

• Graph each piece over its specified interval.

• Mark endpoints with open or closed circles depending on whether the interval includes the endpoint.

Summary Table: Function Properties

Concept	Definition	Example
Relation	Connection between two sets of values	$(2, 5)$
Function	Each input has one output	$f(x) = x^2$
Domain	Set of allowed x-values	$[-4, 5)$
Range	Set of allowed y-values	$[-1, 2)$
Piecewise Function	Function defined by multiple formulas over intervals	$f(x) = \begin{cases} x+1 & x < 0 \\ x^2 & x \geq 0 \end{cases}$

Additional info: These foundational concepts are essential for understanding limits, continuity, and the behavior of functions in calculus.