Lesson 2
Multivariable Functions
Lesson Introduction

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.

What Are Multivariable Functions?

A multivariable function involves more than one input variable. For example, the function f(x,y)=x2+y2f(x, y) = x^2 + y^2 takes two inputs, xx and yy, and produces an output. Here, both xx and yy influence the result of f.

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

f(1,2)=12+22=1+4=5f(1, 2) = 1^2 + 2^2 = 1 + 4 = 5

This function adds the squares of xx and yy to get the result.

Real-Life Example of Multivariable Functions

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:

f(meal cost,tip rate,tax rate)=meal cost×(1+tip rate+tax rate)f(\text{meal cost}, \text{tip rate}, \text{tax rate}) = \text{meal cost} \times (1 + \text{tip rate} + \text{tax rate})

To determine the total cost. For example:

f(50,0.15,0.08)=50×(1+0.15+0.08)=50×1.23=61.5f(50, 0.15, 0.08) = 50 \times (1 + 0.15 + 0.08) = 50 \times 1.23 = 61.5

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

Defining Multivariable Functions in Python

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

Python
1# Defining a multivariable function 2def multivariable_function(x, y): 3 return x**2 + y**2 4 5# Evaluate function at (1, 2) 6result = multivariable_function(1, 2) 7print("f(1, 2) =", result) # f(1, 2) = 5

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

Step-by-Step Explanation
  1. Function Definition: def multivariable_function(x, y): defines a function named multivariable_function that takes two arguments, xx and yy.
  2. Return Statement: return x**2 + y**2 calculates the sum of the squares of xx and yy.
  3. Evaluation: result = multivariable_function(1, 2) evaluates the function with x=1x = 1 and y=2y = 2.
  4. Printing the Result: print("f(1, 2) =", result) prints the output, which is 5.
Exploring More Complex Multivariable Functions

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

f(x,y)=x2xy+y2f(x, y) = x^2 - xy + y^2

Here is how you can define and evaluate it in Python:

Python
1# Defining a more complex multivariable function 2def complex_function(x, y): 3 return x**2 - x*y + y**2 4 5# Evaluate function at (1, 2) 6result = complex_function(1, 2) 7print("f(1, 2) =", result) # f(1, 2) = 3

Just like before, this code defines complex_function and evaluates it at (x=1,y=2)(x = 1, y = 2), resulting in 3.

Visualizing Multivariable Functions

Visualizing helps understand how the function behaves with different inputs. For example, let's plot f(x,y)=x2xy+y2f(x, y) = x^2 - xy + y^2.

The f(x)f(x) is a surface defined by xx and yy. Each possible combination of xx and yy represent a point of this surface.

Lesson Summary

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!

Enjoy this lesson? Now it's time to practice with Cosmo!
Practice is how you turn knowledge into actual skills.