AOS1 Topic 1: Basics of Functions and Notation (Part 1)

Relations, Domain, and Range

An ordered pair, denoted \((x, y)\), is a pair of elements \(x\) and \(y\) in which

\(x\) is considered to be the first coordinate

\(y\) the second coordinate.

A relation is a set of ordered pairs. The following are examples of relations:

\[ a) S = \{(1, 1), (1, 2), (3, 4), (5, 6)\} \]

\[ b) T = \{(-3, 5), (4, 12), (5, 12), (7, -6)\} \]

Every relation determines two sets:

1. The set of all the first coordinates of the ordered pairs is called the domain.
2. The set of all the second coordinates of the ordered pairs is called the range.

For the above examples:

a) domain of \(S = \{1, 3, 5\}\), range of \(S = \{1, 2, 4, 6\}\)

b) domain of \(T = \{-3, 4, 5, 7\}\), range of \(T = \{5, 12, -6\}\)

Some relations may be defined by a rule relating the elements in the domain to their corresponding elements in the range. In order to define the relation fully, we need to specify both the rule and the domain. For example, the set

\[ \{(x, y) : y = x + 1, x \in \{1, 2, 3, 4\}\} \]

is the relation

\[ \{(1, 2),(2, 3), (3, 4), (4, 5)\} \]

The domain is the set \(X = \{1, 2, 3, 4\}\) and the range is the set \(Y = \{2, 3, 4, 5\}\).

When the domain of a relation is not explicitly stated, it is understood to consist of all real numbers for which the defining rule has meaning. For example:

1. \( S = \{ (x, y) : y = x^2 \} \) is assumed to have domain \(\mathbb{R}\)

1. \( T = \{ (x, y) : y = \sqrt{x} \} \) is assumed to have domain \([0, \infty)\)

Basics of Functions:

A function is a rule that assigns each input from a set (domain) to exactly one output from another set (codomain). Functions are widely used in mathematics to model relationships between quantities, transformations, and various other phenomena.

Function Notation:

1. Function Definition: A function is typically defined using the notation \( f(x) \), where \( f \) is the name of the function and \( x \) is the input variable.
2. \[f(x)=2x+3\]
3. 
   
4. Domain and Codomain: The domain is the set of all possible inputs, and the codomain is the set of all possible outputs.
5. 
   
6. Function Evaluation: To evaluate a function at a specific value of \( x \), substitute that value into the function expression.
7. \[f(2)=2(2)+3=7\]
8. 
   
9. Graphing Functions: Functions can be graphed on coordinate axes to represent relationships between inputs and outputs.
10. 
    
11. Function Names: Functions are often named using letters such as \( f \), \( g \), \( h \), etc.
12. 
    
13. Independent and Dependent Variables: The input variable (\( x \)) is the independent variable, and the output variable (\( y \) or \( f(x) \)) is the dependent variable.
14. 
    
15. Function Expressions: Functions are expressed algebraically using formulas or rules that define how inputs are mapped to outputs.

Identifying Functions with the Vertical-Line Test

One way to identify whether a relation is a function is to draw a graph of the relation and then apply the following test:

Vertical-Line Test

Rule: If any vertical line intersects the graph of the relation more than once, the relation is not a function.

Reason: A function assigns exactly one output for each input. Multiple intersections imply multiple outputs for a single input.

The GeoGebra simulation below shows this with a function and a relation

Created with GeoGebra ® , by Dr Wig, Link

Example:

Plot the following graphs on Desmos

Consider the relation $R$ defined by the equation $y=x^2$.

Consider the relation $S$ defined by $x^2+y^2=1$.

If we did a vertical line test on both, the graph of $S$ will fail the test while $R$ will pass. Therefore $S$ is a relation and $R$ is a function

Types of Functions and Implied Domains

1. One-to-One Functions

A function is said to be one-to-one if different \( x \)-values map to different \( y \)-values. That is, a function \( f \) is one-to-one if \( a \neq b \) implies \( f(a) \neq f(b) \), for all \( a, b \in \text{dom} \, f \).

An equivalent way to say this is that a function \( f \) is one-to-one if \( f(a) = f(b) \) implies \( a = b \), for all \( a, b \in \text{dom} \, f \).

Real-World Analogy:

Think about assigning each student in a class to a unique locker. If no two students share the same locker, then the assignment is one-to-one. Each student (input) has their own unique locker (output), and if two students had the same locker, they would actually be the same student.

Example:

The function \( f(x) = 2x + 1 \) is one-to-one because:

\( f(a) = f(b) \implies 2a + 1 = 2b + 1 \)

\( \implies 2a = 2b \)

\( \implies a = b \)

The function \( f(x) = x^2 \) is not one-to-one as, for example, \( f(3) = 9 = f(-3) \).

The vertical-line test can be used to determine whether a relation is a function or not. Similarly, there is a geometric test that determines whether a function is one-to-one or not.

Horizontal-line test

Similar to the Vertical-Line Test, the Horizontal-Line Test helps determine if a function is one-to-one.

Rule: If any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Reason: A one-to-one function cannot have two different inputs producing the same output.

Created with GeoGebra ® , by Irina Boyadzhiev, Link

Example:

Plot the following graphs on Desmos

Consider the function $A(x)$ defined by the equation $A(x)=2x+3$.

Consider the function $B(x)$ defined by $B(x)=x^2$.

If we did a horizontal line test on both, the graph of $A(x)$ will pass the test while $B(x)$ will fail. Therefore $A(x)$ is one to one function.

2. Onto (Surjective) Functions

Definition: A function \( f \) is onto if for every element \( b \) in the codomain, there exists an element \( a \) in the domain such that \( f(a) = b \).

Example: \( f(x) = x^3 \), where the codomain is all real numbers.

Practical Example:

Domain: All the students in your class.

Codomain: All the available project topics.

Onto Function: Every project topic is chosen by at least one student. No topic is ignored.

Non-Onto Example: If there's a project topic that no student picks, then it's not an onto function.

3. Bijective Functions

Definition: A function \( f \) is bijective if it is both one-to-one and onto. This means each element in the codomain is mapped by exactly one element in the domain.

Example: \( f(x) = 2x + 3 \) (if the codomain is also all real numbers).

Real-World Analogy:

Imagine you’re in charge of a seating plan in a classroom. You have the same number of students as there are seats.

1. Domain (Inputs): All the students in the class.
2. Codomain (Possible Outputs): All the available seats.

One-to-One (Injective):

1. Each Student Gets a Unique Seat: No two students share the same seat. If Student A is in Seat 1, then Student B can’t be in Seat 1. Different students have different seats.

Onto (Surjective):

1. All Seats Are Filled: Every seat has a student assigned to it. There are no empty seats left out.

Implied Domains

If the domain of a function is not specified, then the domain is the largest subset of \(\mathbb{R}\) for which the rule is defined; this is called the implied domain or the maximal domain.

Thus, for the function \( f(x) = \sqrt{x} \), the implied domain is \([0, \infty)\). We write:

\( f : [0, \infty) \to \mathbb{R}, \, f(x) = \sqrt{x} \)

This is because $x$ cannot be negative (unless you're doing Specialist Maths...)

Piecewise-defined Functions

Functions which have different rules for different subsets of their domain are called piecewise-defined functions. They are also known as hybrid functions.

An example of this is tax rates. Depending on your income in Australia you get a different tax rate. If $x$ is your income then $Tax(x)$ will be your tax.

Mathematical Representation:

Odd and Even Functions

Odd Functions

An odd function has the property that \( f(-x) = -f(x) \). The graph of an odd function has rotational symmetry with respect to the origin:

The graph remains unchanged after rotation of 180° about the origin.

For example, \( f(x) = x^3 - x \) is an odd function, since

\( f(-x) = (-x)^3 - (-x) \)

\( = -x^3 + x \)

\( = -f(x) \)

Even Functions

An even function has the property that \( f(-x) = f(x) \).

The graph of an even function is symmetrical about the y-axis.

For example, \( f(x) = x^2 - 1 \) is an even function, since

\( f(-x) = (-x)^2 - 1 \)

\( = x^2 - 1 \)

\( = f(x) \)

Determine which function below is even and which is odd.

Created with GeoGebra ® , by April Sur, Link

Key Concepts in Functions:

One-to-One and Many-to-One: A function is one-to-one if each input corresponds to a unique output. It is many-to-one if multiple inputs can produce the same output.

Onto and Into: A function is onto if every element in the codomain is mapped to. It is into if there are elements not mapped to in the codomain.

Inverse Functions: An inverse function \( f^{-1}(y) \) "undoes" the action of the original function, mapping outputs back to inputs.

Composite Functions: Formed by chaining two or more functions together, where the output of one function becomes the input of another.

Piecewise Functions: Have different rules for different parts of their domain, often defined using different expressions for specific intervals or conditions.