Learn Implementing Basic Functions in Python | Functions and Their Properties

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.


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# 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.


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# 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.


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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.


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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.


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# 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

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