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 $$f(4)$$?


This course distills the essentials of Python's math module, focusing on trigonometric functions, logarithmic operations, and mathematical constants. Learners will gain hands-on experience using Python to perform these standard mathematical computations, ideal for beginners seeking practical skills with the math module.

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)