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 and \(y\) 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:

• The set of all the first coordinates of the ordered pairs is called the domain.
• 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:

• \( S = \{ (x, y) : y = x^2 \} \) is assumed to have domain \(\mathbb{R}\)
• \( T = \{ (x, y) : y = \sqrt{x} \} \) 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. Domain and Codomain: The domain is the set of all possible inputs, and the codomain is the set of all possible outputs.
3. Function Evaluation: To evaluate a function at a specific value of \( x \), substitute that value into the function expression.
4. Graphing Functions: Functions can be graphed on coordinate axes to represent relationships between inputs and outputs.

Function Notation:

1. Function Names: Functions are often named using letters such as \( f \), \( g \), \( h \), etc.
2. Independent and Dependent Variables: The input variable (\( x \)) is the independent variable, and the output variable (\( y \) or \( f(x) \)) is the dependent variable.
3. 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

If a vertical line can be drawn anywhere on the graph and it only ever intersects the graph a maximum of once, then the relation is a function.

For example:

Types of Functions and Implied Domains

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 \( a, b \in \text{dom} \, f \).

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

If a horizontal line can be drawn anywhere on the graph of a function and it only ever intersects the graph a maximum of once, then the function is one-to-one.

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.

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).

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:

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

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.

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

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

\( = x^2 - 1 \)

\( = f(x) \)

Key Concepts in Functions:

1. 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.
2. 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.
3. Inverse Functions: An inverse function \( f^{-1}(y) \) "undoes" the action of the original function, mapping outputs back to inputs.
4. Composite Functions: Formed by chaining two or more functions together, where the output of one function becomes the input of another.
5. Piecewise Functions: Have different rules for different parts of their domain, often defined using different expressions for specific intervals or conditions.