Graphical relations and functions | Functions and their graphs | Algebra II | Khan Academy

TL;DR

Determine if the points on a graph represent a function by checking if each input value has only one corresponding output value.

Transcript

Determine whether the points on this graph represent a function. Now, just as a refresher, a function is really just an association between members of a set that we call the domain and members of the set that we call a range. So if I take any member of the domain, let's call that x, and I give it to the function, the function should tell me what me... Read More

Key Insights

• 🔠 A function is an association between input and output values, where each input has a unique output value.
• 📈 To determine if a graph represents a function, check if every input value has only one corresponding output value.
• 📈 A graph is not a function if any input value is associated with multiple output values.
• 🫥 The vertical line test can be used to check if a graph represents a function.
• 🔂 Functions are considered well-behaved relations where each input has a single output.
• 💄 Relations can have multiple outputs for a single input, making them different from functions.
• 🧡 The domain represents the valid input values for a function, and the range represents the corresponding output values.

Questions & Answers

Q: What defines a function?

A function is defined as an association between the members of a domain (input values) and the members of a range (output values), where each input has only one output.

Q: How can we determine if a graph represents a function?

By checking if each input value in the domain has a unique output value in the range. If any input value has multiple associated output values, the graph does not represent a function.

Q: Is it possible for a graph to represent both functions and relations?

Yes, a graph can represent both functions and relations. If each input value has a single corresponding output value, it is a function. If there are points where one input value has multiple output values, it is a relation but not a function.

Q: What is the vertical line test?

The vertical line test is a test used to determine if a graph represents a function. If a vertical line intersects the graph at only one point for each input value, it is a function. If it intersects at multiple points, it is not a function.

Summary & Key Takeaways

• A function is an association between the members of a domain and the members of a range, where each input value has a unique output value.

• For a graph to represent a function, each input value in the domain must have only one corresponding output value in the range.

• If a graph has any points where multiple output values are associated with a single input value, it is not a function but a relation.