Definition and Types of Functions (One-to-One, Onto, etc.)

Introduction

Key Concepts

1. Definition of Functions

A function is a relation between two sets that assigns each element of the first set, called the domain, to exactly one element of the second set, known as the codomain. Formally, a function \( f \) from set \( A \) to set \( B \) is denoted as \( f: A \rightarrow B \), where for every \( a \in A \), there exists a unique \( b \in B \) such that \( f(a) = b \).

**Example:** Consider the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) defined by \( f(x) = 2x + 3 \). Here, each real number \( x \) is mapped to another real number \( 2x + 3 \).

2. One-to-One (Injective) Functions

A function \( f: A \rightarrow B \) is called **one-to-one** or **injective** if different elements in the domain \( A \) map to different elements in the codomain \( B \). In other words, if \( f(a_1) = f(a_2) \), then \( a_1 = a_2 \).

Mathematically, \( f \) is injective if: $$ \forall a_1, a_2 \in A, \ f(a_1) = f(a_2) \implies a_1 = a_2 $$

**Example:** The function \( f(x) = 2x + 3 \) is injective because if \( 2x_1 + 3 = 2x_2 + 3 \), then \( x_1 = x_2 \).

3. Onto (Surjective) Functions

A function \( f: A \rightarrow B \) is called **onto** or **surjective** if every element in the codomain \( B \) has at least one corresponding element in the domain \( A \). Formally, for every \( b \in B \), there exists at least one \( a \in A \) such that \( f(a) = b \).

Mathematically, \( f \) is surjective if: $$ \forall b \in B, \ \exists a \in A \text{ such that } f(a) = b $$

**Example:** The function \( f(x) = x^2 \) from \( \mathbb{R} \) to \( \mathbb{R} \) is not surjective since negative numbers in \( \mathbb{R} \) are not outputs of \( f \). However, if the codomain is restricted to \( \mathbb{R}_{\geq 0} \), then \( f \) becomes surjective.

4. Bijective Functions

A function \( f: A \rightarrow B \) is called **bijective** if it is both injective and surjective. This means every element in the domain maps to a unique element in the codomain, and every element in the codomain is mapped by some element in the domain.

Mathematically, \( f \) is bijective if: $$ f \text{ is injective and surjective} $$

**Example:** The function \( f(x) = 2x + 3 \) from \( \mathbb{R} \) to \( \mathbb{R} \) is bijective because it is both injective and surjective.

5. Even and Odd Functions

Functions can also be classified based on their symmetry properties:

• Even Function: A function \( f \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain.
• Odd Function: A function \( f \) is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.

**Example of Even Function:** \( f(x) = x^2 \). Since \( (-x)^2 = x^2 \).

**Example of Odd Function:** \( f(x) = x^3 \). Since \( (-x)^3 = -x^3 \).

6. Composite Functions

A **composite function** is formed when one function is applied to the result of another function. If \( f: A \rightarrow B \) and \( g: B \rightarrow C \), then the composite function \( g \circ f: A \rightarrow C \) is defined by: $$ (g \circ f)(a) = g(f(a)) \quad \forall a \in A $$

**Example:** Let \( f(x) = 2x + 3 \) and \( g(x) = x^2 \). Then, $$ (g \circ f)(x) = g(f(x)) = (2x + 3)^2 $$

7. Inverse Functions

An **inverse function** reverses the effect of the original function. If \( f: A \rightarrow B \) is bijective, then its inverse \( f^{-1}: B \rightarrow A \) satisfies: $$ f^{-1}(f(a)) = a \quad \forall a \in A $$ $$ f(f^{-1}(b)) = b \quad \forall b \in B $$

**Example:** If \( f(x) = 2x + 3 \), then to find \( f^{-1}(x) \): \begin{align*} y &= 2x + 3 \\ y - 3 &= 2x \\ x &= \frac{y - 3}{2} \\ \implies f^{-1}(x) &= \frac{x - 3}{2} \end{align*}

Advanced Concepts

1. Mathematical Proofs of Injectivity and Surjectivity

Understanding the properties of injective and surjective functions often requires formal mathematical proofs. Let's explore proofs for both properties using different functions.

Proof of Injectivity

**Theorem:** The function \( f(x) = 3x + 2 \) is injective.

**Proof:** Assume \( f(x_1) = f(x_2) \). \begin{align*} 3x_1 + 2 &= 3x_2 + 2 \\ 3x_1 &= 3x_2 \\ x_1 &= x_2 \end{align*} Since \( x_1 = x_2 \), the function \( f \) is injective.

Proof of Surjectivity

**Theorem:** The function \( f(x) = x^3 \) is surjective when considered from \( \mathbb{R} \) to \( \mathbb{R} \).

**Proof:** Let \( y \) be any real number. We need to find an \( x \in \mathbb{R} \) such that \( f(x) = y \). \begin{align*} y &= x^3 \\ x &= \sqrt[3]{y} \end{align*} Since \( \sqrt[3]{y} \) is real for all \( y \in \mathbb{R} \), such an \( x \) exists. Therefore, \( f \) is surjective.

2. Cardinality and Bijective Functions

Cardinality refers to the number of elements in a set. When two sets have the same cardinality, there exists a bijective function between them. This concept is pivotal in understanding the sizes of infinite sets.

**Example:** Consider the sets \( \mathbb{N} \) (natural numbers) and \( \mathbb{Z} \) (integers). Both are countably infinite, and there exists a bijective function between them, demonstrating that their cardinalities are equal.

**Implication:** If there exists a bijection between sets \( A \) and \( B \), then \( |A| = |B| \), where \( |A| \) denotes the cardinality of \( A \).

3. Composition and Inversion of Functions

When working with composite functions and their inverses, certain properties can be derived that are essential for simplifying complex expressions.

**Property:** If \( f \) and \( g \) are bijective functions, then \( (g \circ f)^{-1} = f^{-1} \circ g^{-1} \).

**Proof:** \begin{align*} (g \circ f)^{-1}(y) &= f^{-1}(g^{-1}(y)) \end{align*} This shows that the inverse of the composite function \( g \circ f \) is the composite of the inverses in reverse order.

4. Interdisciplinary Applications

Functions play a critical role in various fields beyond mathematics, such as physics, engineering, and economics. Understanding the types of functions enhances their application in modeling real-world phenomena.

**Physics Example:** In kinematics, the position of an object as a function of time is often modeled using linear or quadratic functions, depending on the motion's nature.

**Economics Example:** Supply and demand curves are represented as functions, where understanding their injectivity and surjectivity can determine market equilibria.

5. Advanced Problem Solving

To master functions, students must engage in complex problem-solving that integrates multiple concepts. Consider the following problem:

Problem: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a bijective function defined by \( f(x) = ax + b \), where \( a \neq 0 \). Find the inverse function \( f^{-1}(x) \) and prove its bijectivity.

Solution:

To find the inverse function, solve for \( x \): \begin{align*} y &= ax + b \\ y - b &= ax \\ x &= \frac{y - b}{a} \\ \implies f^{-1}(x) &= \frac{x - b}{a} \end{align*}

**Proof of Bijectivity:** \begin{itemize>

Since \( f^{-1} \) is both injective and surjective, it is bijective.

6. Function Composition in Polynomial Functions

Consider the composition of two polynomial functions:

Let \( f(x) = x^2 + 1 \) and \( g(x) = 3x - 2 \). The composite function \( h = g \circ f \) is: $$ h(x) = g(f(x)) = 3(x^2 + 1) - 2 = 3x^2 + 1 $$

Analyzing the properties:

• Injectivity: Since \( h(x) = 3x^2 + 1 \) is not injective over \( \mathbb{R} \) (both \( x \) and \( -x \) yield the same output), it is not injective.
• Surjectivity: The range of \( h \) is \( [1, \infty) \), so if the codomain is \( \mathbb{R} \), \( h \) is not surjective. However, if the codomain is \( [1, \infty) \), then \( h \) is surjective.

Thus, \( h(x) \) is not bijective when considered from \( \mathbb{R} \) to \( \mathbb{R} \), but it is injective and surjective when the codomain is appropriately restricted.

7. Applications in Calculus: Inverse Functions and Derivatives

Inverse functions are essential in calculus, especially when dealing with derivatives. The derivative of an inverse function can be expressed in terms of the derivative of the original function.

**Formula:** If \( f \) is differentiable and bijective with inverse \( f^{-1} \), then: $$ (f^{-1})'(y) = \frac{1}{f'(x)} \quad \text{where} \ x = f^{-1}(y) $$

**Example:** Let \( f(x) = 2x + 3 \). Its inverse is \( f^{-1}(x) = \frac{x - 3}{2} \). \begin{align*} f'(x) &= 2 \\ (f^{-1})'(y) &= \frac{1}{2} \end{align*}

This relationship simplifies the computation of derivatives for inverse functions without directly differentiating the inverse function itself.

8. Exploring Non-Bijective Functions

Not all functions are bijective. Understanding the limitations and characteristics of non-bijective functions is crucial for comprehending the broader landscape of function properties.

**Example:** The function \( f(x) = x^2 \) from \( \mathbb{R} \) to \( \mathbb{R} \) is neither injective nor surjective. It fails to be injective because \( f(a) = f(-a) \) for any \( a \neq 0 \), and it fails to be surjective since negative real numbers are not in the range.

**Implications:** Non-bijective functions do not have inverses over their entire domain and codomain. However, by restricting the domain or codomain, certain non-bijective functions can become bijective.

Comparison Table

Summary and Key Takeaways