Multivariable Functions

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 $y$, and produces an output. Here, both $x$ and $y$ influence the result of f.

Suppose you have $x = 1$ and $y = 2$:

This function adds the squares of $x$ and $y$ to get the result.

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:

To determine the total cost. For example:

This function considers the meal's cost, tip, and tax rate to calculate the final amount you'll pay.

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:

# Defining a multivariable function
def multivariable_function(x, y):
    return x**2 + y**2

# Evaluate function at (1, 2)
result = multivariable_function(1, 2)
print("f(1, 2) =", result)  # f(1, 2) = 5

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