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 takes two inputs, and , and produces an output. Here, both and influence the result of f
.
Suppose you have and :
This function adds the squares of and to get the result.
Consider calculating the total cost of a meal where the tip and tax vary based on different percentages:
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, and , and returns their sum of squares:
Python1# 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 , resulting in 5.
def multivariable_function(x, y):
defines a function named multivariable_function
that takes two arguments, and .return x**2 + y**2
calculates the sum of the squares of and .result = multivariable_function(1, 2)
evaluates the function with and .print("f(1, 2) =", result)
prints the output, which is 5.Functions in machine learning can get complex. Consider a surface modeling function:
Here is how you can define and evaluate it in Python:
Python1# 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 , resulting in 3.
Visualizing helps understand how the function behaves with different inputs. For example, let's plot .
The is a surface defined by and . Each possible combination of and represent a point of this surface.
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!