Functions, Relations, Domain, and Range

Welcome back to Professor Baker's Math Class! Today, we explored the fundamental concepts of functions, relations, domain, and range. Understanding these concepts is crucial for success in algebra and beyond. Let's review what we covered:

What is a Relation?

A relation is simply a connection or correspondence between two sets of data. Think of it as a way to pair elements from one set to another. For example, a relation could describe the relationship between students and their favorite colors, or between cities and their time zones.

Here's an example of a relation:

  • Jordan → Black
  • Sam → Grey
  • Jen → Blue

What is a Function?

A function is a special type of relation where each input (from the domain) has exactly one output (in the range). This is often described as a "one-to-one" relationship, but it's more accurate to say each input has only one output, even if different inputs can share the same output.

Key Idea: To determine if a relation is a function, check if any input has more than one output. If it does, it's a relation but not a function.

Domain and Range

  • Domain: The set of all possible input values (often called the 'x' values).
  • Range: The set of all possible output values (often called the 'y' values).

In mathematical notation, we often use $x$ to represent the input and $y$ to represent the output. A function can be written as $y = f(x)$, where $f$ is the function name.

Examples

Let's look at some examples to solidify our understanding. Consider the following relation represented as a set of ordered pairs: {(1, -2), (2, -4), (3, -8)}.

  • Domain: {1, 2, 3}
  • Range: {-2, -4, -8}
  • Is it a function? Yes, because each input (1, 2, and 3) has only one output.

Now, let's consider this relation: {(-3, 3), (1, -2), (1, 1), (4, 4)}

  • Domain: {-3, 1, 4}
  • Range: {3, -2, 1, 4}
  • Is it a function? No! The input '1' has two outputs (-2 and 1). This violates the rule for functions.

Homework

Time to practice! Please complete the following exercises to reinforce your understanding of functions, relations, domain, and range:

  1. Pg. 71 #1-3
  2. Pg. 71 #19-21
  3. Write down a relation that is a function and one that is not a function. Explain why each one fits (or doesn't fit) the definition of a function.

Keep practicing, and remember: every input in a function gets only one output! You've got this!