Algebra 16 - Real-Valued Functions of a Real Variable

Summary

In this lecture, Professor Von Schmohawk introduces the concept of functions as special types of binary relations, where each input from the domain is associated with exactly one output in the range. The function is defined by a condition, often expressed through an equation, such as y = 2x. The lecture explains how functions can be represented graphically and formally using set notation, with a focus on the function’s domain, codomain, and range. The difference between independent and dependent variables is clarified, and the importance of defining the domain in mathematical functions is emphasized.

Takeaways

• 😀 Functions are a type of binary relation that map elements from a set called the 'domain' to a set called the 'range'.
• 😀 A function is defined as a set of ordered pairs, where the first element comes from the domain and the second from the range.
• 😀 In algebra, functions often involve real numbers, and their graphs can be visualized as points on the Cartesian plane.
• 😀 If a function has an infinite number of ordered pairs, it can be defined using a mathematical condition or equation, rather than listing each pair.
• 😀 An example of a function is one where the output (y) is always twice the input (x), represented by the equation 'y = 2x'.
• 😀 Functions can be formally defined using set notation, such as 'f = {(x, y) | x ∈ R, y = 2x}', where the domain is the set of real numbers (R).
• 😀 The notation 'f(x) = 2x' is an alternative, more common way of expressing the same function.
• 😀 Functions are often described with an equation in algebra, where variables represent real numbers unless otherwise stated.
• 😀 In the equation 'y = 2x', x is the 'independent variable' (since it can take any real value), and y is the 'dependent variable' (since its value depends on x).
• 😀 The domain and range of a function with real numbers as inputs and outputs are both typically assumed to be the set of real numbers (R).
• 😀 The notation 'f: R → R' indicates a function from the real numbers (R) to the real numbers (R), also called a real-valued function of a real variable.

Q & A

What is a function in mathematics?

A function is a special type of binary relation that associates each element from a set of inputs (called the domain) to an element in a set of outputs (called the range).

How are functions typically represented in algebra?

Functions are often represented as ordered pairs, where the first element of each pair is from the domain and the second element is from the range. In algebra, these pairs are typically real numbers, and the graph of the function is plotted on the Cartesian plane.

What is the problem with defining functions with an infinite number of ordered pairs?

Defining a function with an infinite number of ordered pairs is impractical. Instead of listing all pairs, we can define a condition or equation that describes all the pairs in the function, such as a relationship between the input and output values.

How can we define a function that has an infinite number of ordered pairs?

A function with an infinite number of ordered pairs can be defined by stating a condition that each pair must satisfy, usually expressed mathematically as an equation. For example, for the graph of a line, the condition might be y = 2x.

What does the equation y = 2x describe in terms of a function?

The equation y = 2x describes a function where the output (y) is always twice the value of the input (x). It represents a set of ordered pairs where the second element is twice the first element.

What is the formal set notation for a function, and what does it represent?

The formal set notation for a function is written as { (x, y) | x ∈ R, y = 2x }, which represents the set of all ordered pairs where x is a real number, and y is twice the value of x. This defines the function formally by its set of input-output pairs.

What is the difference between function notation (f(x)) and set notation?

Function notation, such as f(x), represents the output of the function for a given input x. Set notation, on the other hand, formally defines the function as a set of ordered pairs, specifying the relationship between inputs and outputs, often with conditions like y = 2x.

What are independent and dependent variables in the context of a function?

In a function, the independent variable is the input (x), which can take any value from the domain. The dependent variable is the output (y), which depends on the value of the independent variable (x). In the example y = 2x, x is the independent variable and y is the dependent variable.

What does it mean when we say that the domain and range of a function are real numbers?

When we say that the domain and range of a function are real numbers, it means that both the input values (x) and output values (y) can be any real number. In the case of the function y = 2x, both x and y can take any real value.

What is a codomain, and how does it relate to the range of a function?

The codomain of a function is the set of all possible output values that could theoretically be produced by the function. The range is the actual set of output values that the function produces. In many cases, the codomain and range are the same, especially when dealing with functions like f(x) = 2x, where both are real numbers.