Relations & Functions

Summary

This video explains the concepts of relations and functions, emphasizing that a function must have unique X values in the domain with no repeats. It demonstrates how to identify functions using ordered pairs, listing the domain and range, and creating mapping diagrams. Additionally, the video introduces the vertical line test, which helps determine if a graph represents a function. If a vertical line intersects the graph at more than one point, it's not a function. Examples show how to apply these methods to both ordered pairs and graphs, helping viewers better understand the topic.

Takeaways

• 😀 Functions are relations where each X-value has a unique Y-value (no repeating X-values).
• 😀 To determine if a relation is a function, check if any X-values repeat in ordered pairs.
• 😀 The vertical line test is used to determine if a graph represents a function—if a vertical line touches the graph at more than one point, it's not a function.
• 😀 A function can have repeating Y-values, but repeating X-values disqualify it as a function.
• 😀 In a function, the domain consists of all unique X-values, and the range consists of all unique Y-values.
• 😀 Mapping diagrams visually represent the relationship between the domain and range by connecting values with arrows.
• 😀 When analyzing ordered pairs, the domain is the set of X-values and the range is the set of Y-values.
• 😀 Even if X-values repeat in the graph's ordered pairs, the vertical line test helps determine if the graph is a function.
• 😀 If a graph passes the vertical line test, it is a function because each X-value is associated with only one Y-value.
• 😀 If a vertical line intersects the graph more than once at any point, the graph is not a function.
• 😀 The video provides examples of both functions and non-functions using both ordered pairs and graphs with the vertical line test.

Q & A

What defines a function in mathematics?

A function is a relation where each X value (input) corresponds to a unique Y value (output). This means no X value is repeated in a function.

How can we determine if a relation is a function using ordered pairs?

To determine if a relation is a function, check the X values in the ordered pairs. If any X value repeats, the relation is not a function.

Is it okay for Y values to repeat in a function?

Yes, it is okay for Y values to repeat in a function as long as the X values are unique.

What is the domain of a function?

The domain of a function is the set of all X values (inputs) that appear in the ordered pairs or are part of the relation.

What is the range of a function?

The range of a function is the set of all Y values (outputs) that are associated with the elements in the domain, with duplicates removed.

How is a mapping diagram created?

A mapping diagram is created by placing the domain and range in separate 'bubbles' and drawing arrows to represent the relationships between each domain element and its corresponding range element.

What does the vertical line test determine?

The vertical line test is used to determine whether a graph represents a function. If a vertical line crosses the graph more than once, the relation is not a function.

Why is it important for a graph to pass the vertical line test?

It is important for a graph to pass the vertical line test because it ensures that each X value has only one corresponding Y value, which is a key characteristic of a function.

What happens if a graph fails the vertical line test?

If a graph fails the vertical line test, it means that a vertical line touches the graph at more than one point, indicating that the relation is not a function.

Can a relation still be a function if the Y values repeat?

Yes, a relation can still be a function if the Y values repeat, as long as the X values do not repeat. The uniqueness of the X values is what defines a function.