MCR3U Introduction to Functions and Key Features

Introduction to Relations and Functions

In mathematical and everyday contexts, many quantities depend on one or more variables that are subject to change. A relation is a rule that describes how these quantities depend on one another.

Everyday Examples of Dependency:
- Speed: Depends on the distance travelled and the time taken.
- Plant Growth: Depends on the duration of exposure to the sun and the amount of water provided.
- Test Marks: Depend on work ethic, attitude, and practice.
- Anatomy: A relationship exists between height and arm span; generally, if a person is taller, their arm span is expected to be longer.

Defining the Function

A function is a specific, "well-behaved" type of relation. For a relation to qualify as a function, every unique input must have exactly one corresponding output.

The Juicer Analogy: Consider an orange juice machine. If you input an orange, you expect only one output: orange juice. If the machine were to output both orange juice and grape juice from a single orange input, it would not be functioning as a "well-behaved" relation (a function).
The Car Analogy: In the context of driving, your location is a function of time. It is impossible to be in two different geographical locations at the exact same moment in time.

Investigation: Comparing $y = x^2$ and $x = y^2$

To differentiate between a general relation and a function, consider the following two equations and their tables of values:

Equation 1: $y = x^2$

Input $(x)$	Output $(y)$	Coordinate $(x, y)$
$-2$	$4$	$(-2, 4)$
$-1$	$1$	$(-1, 1)$
$0$	$0$	$(0, 0)$
$1$	$1$	$(1, 1)$
$2$	$4$	$(2, 4)$

Equation 2: $x = y^2$

Input $(x)$	Output $(y)$	Coordinate $(x, y)$
$4$	$-2$	$(4, -2)$
$1$	$-1$	$(1, -1)$
$0$	$0$	$(0, 0)$
$1$	$1$	$(1, 1)$
$4$	$2$	$(4, 2)$

Observation: In the first equation ( $y = x^2$ ), each unique $x$ has only one $y$ . In the second equation ( $x = y^2$ ), the inputs $1$ and $4$ both result in two separate outputs (e.g., input $4$ yields both $2$ and $-2$ ). Therefore, $y = x^2$ is a function, while $x = y^2$ is a relation but not a function.

The Vertical Line Test (VLT)

The Vertical Line Test is a visual method Used to determine if a graph represents a function.

The Rule: If every vertical line drawn through a graph passes through no more than one point, the relation is a function.
Interpretation:
- If a vertical line passes through two or more points at any x-value, the relation is not a function because that single input has multiple outputs.
- If a vertical line moved from left to right never intersects the graph more than once, the relation is a function.

Modal Representations of Functions

Functions can be modeled in several distinct ways. Consider a catering job where you are paid $\$15/\text{hour}$ . The pay $(P)$ is related to time $(t)$ . This can be shown as follows:

Set Notation: A collection of ordered pairs listed inside curly brackets.
- $\{(0, 0), (1, 15), (2, 30), (3, 45), (4, 60), (5, 75)\}$
Equation: A clear mathematical rule relating the variables.
- $P = 15t$
Table of Values: A structured list of inputs and their corresponding outputs.
Graph: A visual representation of ordered pairs on a Cartesian plane.
Mapping Diagram: A flowchart-style visual showing how x-values are paired with y-values.

Mapping Diagrams and Types of Mappings

Mapping diagrams use arrows to connect input elements to output elements.

One-to-One Mapping: Each input has exactly one unique output. This is a function.
Many-to-One Mapping: Multiple different inputs associate with the same single output. This is still a function (e.g., both $-2$ and $2$ square to result in $4$ ).
One-to-Many Mapping: A single input associates with multiple outputs. This is not a function.

Set Notation and Theory

In mathematics, a set is a collection of elements listed within curly brackets.

Elements: The individual items inside the set (e.g., objects on a desk: $\{\text{computer, mouse, notebook, pencil, eraser, calculator}\}$ ).
Extracting Sets from Points: For the set of points $\{(0, 1), (0, 3), (1, 4), (2, 2), (2, 5)\}$ :
- Set of x-values: $x = \{0, 1, 2\}$
- Set of y-values: $y = \{1, 2, 3, 4, 5\}$

Concepts of Domain and Range

Domain: The set of all possible input values ( $x$ -values) for which the function is defined.
Range: The set of all possible output values ( $y$ -values) resulting from the domain inputs.
Contextual Importance: When determining domain and range, one must consider the real-world context and mathematical restrictions (e.g., a bank account cannot have negative weeks, and a square root cannot be taken of a negative number in the real number system).

Examples of Domain and Range

Bank Account Scenario: You have $\$500$ and spend $\$100$ weekly for $5$ weeks.
- Domain: $D = \{0, 1, 2, 3, 4, 5\}$
- Range: $R = \{0, 100, 200, 300, 400, 500\}$
Square Root Function: $y = \sqrt{x-1}$
- Restriction: The expression under the square root must be non-negative: $x - 1 \ge 0 \rightarrow x \ge 1$ .
- Domain: $D: \{x \in \mathbb{R} \mid x \ge 1\}$
- Range: $R: \{y \in \mathbb{R} \mid y \ge 0\}$
School T-Shirt Order: Initial cost $\$200$ , plus $\$5$ per shirt. Minimum order is $40$ ; maximum budget is $\$1000$ .
- Minimum cost: $5(40) + 200 = 400$
- Maximum shirts: $5x + 200 = 1000 \rightarrow 5x = 800 \rightarrow x = 160$
- Domain: $\text{Number of shirts } x \text{ where } 40 \le x \le 160$
- Range: $\text{Total cost } y \text{ where } 400 \le y \le 1000$

Number Systems Notation

When defining domain and range, specifically in set notation, the following symbols are used to denote number systems:

$\mathbb{I}$ : The set of all Integers.
$\mathbb{Q}$ : The set of all Rational numbers.
$\mathbb{R}$ : The set of all Real numbers.
$\in$ : This symbol means "is an element of."

Standard set notation format: $D: \{x \in \mathbb{R} \mid \text{restriction}\}$ reads as "x belongs to the set of all real numbers such that [restriction exists]."

Function Notation

Function notation allows for the differentiation between multiple functions (e.g., distinguishing between $y = x + 1$ and $y = x^2$ ) by assigning names to functions.

The Syntax: Instead of writing $y = x + 1$ , we write $f(x) = x + 1$ .
Reading notation: $f(x)$ is read as "f of x."
Components:
- $f$ : The name of the function.
- $(x)$ : Signifies that $x$ is the input variable.
- Common identifiers include $f(x)$ , $g(x)$ , and $h(x)$ .

Function Calculations

To find the output for a specific input, substitute the input value into the function equation.

For $f(x) = x + 1$ :
- $f(1) = (1) + 1 = 2$
- $f(-2) = (-2) + 1 = -1$
- $f(3) = (3) + 1 = 4$
For $f(x) = x^2$ :
- $f(1) = (1)^2 = 1$
- $f(-2) = (-2)^2 = 4$
- $f(3) = (3)^2 = 9$
For $f(x) = \sqrt{x}$ :
- $f(1) = \sqrt{1} = 1$
- $f(-2) = \sqrt{-2}$ (Undefined in Real Numbers)
- $f(3) = \sqrt{3} \approx 1.732$

Questions & Discussion

What do you suppose the notation means if $f(2) = -1$ , $f(0) = 3$ , and $f(-2) = 4$ ? This notation indicates that when the input is $2$ , the output is $-1$ ; when the input is $0$ , the output is $3$ ; and when the input is $-2$ , the output is $4$ . These represent the specific ordered pairs $(2, -1)$ , $(0, 3)$ , and $(-2, 4)$ .

Would it be feasible to list all possible points on a continuous function using set notation or tables? No. For functions defined over a range of real numbers (continuous functions), there are an infinite number of points between any two integers. Mapping diagrams, tables, and set notation are generally used for discrete sets of points, whereas graphs and equations are more efficient for continuous functions.