Welcome to our lesson on "Multivariable Functions"! Understanding how functions work with multiple variables is crucial in machine learning, where models depend on more than one parameter. By the end of this lesson, you will understand what multivariable functions are, why they're important, and how to work with them in Python.

Imagine you want to predict the price of a house based on its size and location. Both size and location affect the price. Multivariable functions help us model such relationships.

A multivariable function involves more than one input variable. For example, the function $f(x, y) = x^2 + y^2$ takes two inputs, $x$ and , and produces an output. Here, both and influence the result of .

Consider calculating the total cost of a meal where the tip and tax vary based on different percentages:

• Meal cost: $50
• Tip rate: 15%
• Tax rate: 8%

A multivariable function could be written as:

Let's define and use multivariable functions in Python. Start with a simple function that takes two parameters, $x$ and $y$, and returns their sum of squares:

This example defines the multivariable_function and evaluates it at $(x = 1, y = 2)$, resulting in 5.

1. Function Definition: def multivariable_function(x, y): defines a function named multivariable_function that takes two arguments, $x$ and $y$.
2. Return Statement: return x**2 + y**2 calculates the sum of the squares of $x$ and $y$.
3. Evaluation: result = multivariable_function(1, 2) evaluates the function with and .

Functions in machine learning can get complex. Consider a surface modeling function:

Visualizing helps understand how the function behaves with different inputs. For example, let's plot $f(x, y) = x^2 - xy + y^2$.

Congratulations! You've learned what multivariable functions are and why they're important. We talked about how these functions involve multiple inputs and how to implement them in Python. We even visualized a multivariable function to understand its behavior better.

It's time to practice. In the practice session, you'll use your newly acquired skills to create, evaluate, and understand more complex multivariable functions. Let's get coding!