Summary  
This chapter demonstrates how to define and implement code for various mathematical function types in Python—showing how inputs map to outputs in one-to-one, many-to-one, onto, into, and bijective functions by printing their outputs.

General domain of usage  
Mathematics

Functions define relationships between inputs and outputs, making them fundamental in mathematics, programming, and data science. In Python, we can define and visualize different types of functions, such as **one-to-one**, **many-to-one**, **onto**, **into**, and **bijective functions**.

## Types of Functions in Python

### One-to-One (Injective) Function

A **one-to-one function** ensures that each input maps to a **unique** output. As you'll see, no two inputs have the same output.


# One-to-One Function: f(x) = x
def one_to_one(x):
    return x  
    
# Example Outputs
print("One-to-One Function Outputs:")
print(one_to_one(2))    # Output is 2
print(one_to_one(5))    # Output is 5


### Many-to-One Function
A **many-to-one function** allows **multiple** inputs to map to the **same** output.

# Many-to-One Function: f(x) = x^2
def many_to_one(x):
    return x ** 2  
    
# Example Outputs
print("\nMany-to-One Function Outputs:")
print(many_to_one(3))    # Output is 9  
print(many_to_one(-3))   # Output is also 9 (Same output for different inputs) 

### Onto (Surjective) Function
An **onto function** ensures that **every possible output in the codomain** has at least **one** input mapped to it.

import numpy as np

# Onto Function: f(x) = tan(x)
def onto(x):
    return np.tan(x)  
    
# Example Outputs
print("\nOnto Function Outputs:")
print(onto(1))    # Output is approximately 1.557
print(onto(-1))   # Output is approximately -2.185


### Into Function
An into function means not all values in the codomain are covered—some outputs remain unused.

import numpy as np

# Into Function: f(x) = sin(x) (Only outputs between -1 and 1)
def into(x):
    return np.sin(x)  

# Example Outputs
print("\nInto Function Outputs:")
print(into(0))            # Output is approximately 0
print(into(np.pi / 2))    # Output is approximately 1

### Bijective Function (One-to-One & Onto)
A **bijective function** is **both one-to-one and onto**, meaning it is **invertible**.

# Bijective Function: f(x) = x
def bijective(x):
    return x  
    
# Example Outputs
print("\nBijective Function Outputs:")
print(bijective(3))    # Output is 3
print(bijective(-4))   # Output is -4

What will the following function return for $$f(4)$$?


Master the mathematical foundations essential for data science. Explore core concepts in functions, calculus, linear algebra, probability, and dimensionality reduction. Build both theoretical understanding and practical coding experience to strengthen your ability to analyze data, model complex systems, and apply advanced techniques in machine learning.

Explore the foundation of mathematical functions. Learn different types of algebraic and transcendental functions, their properties, and how to implement them in Python to solve real-world problems.

Master the concepts of sets and series, from basic operations to practical applications. Gain hands-on experience implementing set operations and working with arithmetic and geometric series in Python.

Develop a solid understanding of limits, derivatives, integrals, and partial derivatives. Connect theory to practice by implementing these concepts in Python and applying them to optimization through gradient descent.

Build strong knowledge of vectors, matrices, and transformations. Learn decomposition methods and eigenvalue analysis, while reinforcing concepts with Python coding challenges and practical data science applications.

Dive into probability theory and statistics. Study conditional probability, Bayes' theorem, and statistical measures. Implement key concepts in Python, simulate distributions, and strengthen your skills through challenges and quizzes.

Implementing Basic Functions in Python

Types of Functions in Python

One-to-One (Injective) Function

Many-to-One Function

Onto (Surjective) Function

Into Function

Bijective Function (One-to-One & Onto)