Study guide: Understanding Relations and Functions

📋 Course Outline

1. Relations and functions fundamentals
2. Function notation, domain and range

📖 1. Relations and functions fundamentals

🔑 Key Concepts & Definitions

• Relation: A relation is a set of ordered pairs that links inputs to outputs.
• Function: A function is a special relation where each input has exactly one output.
• Input-output mapping: An input-output mapping describes how each input is assigned to an output.

📝 Essential Points

• A relation can assign multiple outputs to the same input, while a function cannot.
• To test whether a relation is a function, check that no input appears with two different outputs.
• In ordered pairs, the first coordinate is the input and the second coordinate is the output.

💡 Memory Hook

Function = one input → one output; relation can be many-to-one.

📖 2. Function notation, domain and range

🔑 Key Concepts & Definitions

• Function notation: Function notation writes the output of a function using the form $f(x)$.
• Domain: The domain is the set of all inputs for which the function is defined.
• Range: The range is the set of all outputs the function actually produces.

📝 Essential Points

• If $f(x)$ is defined for a value of $x$, that $x$ belongs to the domain.
• Values of $x$ outside the domain are not allowed in $f(x)$.
• The range is determined by the outputs corresponding to inputs in the domain.

💡 Memory Hook

Domain = allowed x; Range = resulting y.

⚠️ Common Pitfalls & Confusions

1. Students often confuse domain (allowed inputs) with range (produced outputs).
2. Students may treat a relation as a function even when one input has two different outputs.
3. Students sometimes swap the order in ordered pairs, misreading input as output.

✅ Exam Checklist

1. Identify whether a given set of ordered pairs defines a function by checking the “one input → one output” condition.
2. Write and interpret function notation $f(x)$ for specified inputs.
3. Determine the domain from which $x$ values make the function defined.
4. Determine the range from the set of outputs produced by the allowed inputs.
5. Use ordered pairs consistently: first coordinate as input and second as output.