how to apply a function to an numpy array

## Applying Functions to NumPy Arrays: A Comprehensive Guide

NumPy arrays are the foundation of numerical computing in Python. Their efficiency and optimized operations make them ideal for handling large datasets. One common task is applying a function to each element, or more generally, to sections of a NumPy array. This tutorial will guide you through various methods for applying functions to NumPy arrays, along with detailed explanations, examples, and considerations for choosing the right approach.

**1. Element-Wise Application: The Foundation**

The simplest scenario involves applying a function to each individual element of the array. NumPy facilitates this in several ways:

**1.1. Using Vectorization (Best Practice):**

Vectorization is NumPy's superpower. It leverages optimized C code under the hood to perform operations on entire arrays simultaneously, avoiding explicit loops. If your function is amenable to vectorization (i.e., can operate directly on NumPy arrays), this is almost always the most efficient method.

**Explanation:**

* The `square` function takes a single number as input. However, when we pass the entire `arr` array to it, NumPy's vectorization mechanism automatically applies `square` to each element of `arr`, returning a new array `squared_arr` containing the results.

**Key Advantages of Vectorization:**

* **Speed:** Significantly faster than explicit loops, especially for large arrays.
* **Readability:** Code is more concise and easier to understand.
* **Efficiency:** Leverages optimized NumPy routines.

**1.2. NumPy's Universal Functions (ufuncs):**

NumPy provides a wealth of built-in functions called "universal functions" (ufuncs) designed for element-wise operations. These ufuncs are highly optimized and are the preferred way to perform common mathematical operations.

**Important NumPy ufuncs:**

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