Graphs of Functions and Non-Functions

Understanding Functions and Non-Functions in Graphs

• A function is a relationship where each input has exactly one output.
• In a function graph, any vertical line drawn will intersect the graph at most once, representing a single y-value for each x-value.
• Not a function indicates a relationship where at least one input has multiple outputs, violating the strict definition of a function.

Characteristics of Functions

• Functions can be represented as equations, graphs, or tables.
• Example forms of functions include linear, quadratic, polynomial, and exponential.
• Functions may have domains and ranges that influence their behavior and characteristics.

Identifying Non-Functions

• A graph shows as not a function if it fails the vertical line test, meaning a vertical line intersects the graph at two or more points.
• Common examples of non-functions include circles and parabolas opening sideways.
• Non-functions can also be described through multi-valued relationships, such as square roots or absolute values that produce multiple outputs for a single input.

Types of Functions

• Linear Functions: Represented by a straight line; has a constant slope.
• Quadratic Functions: Form a parabola and can open upwards or downwards.
• Exponential Functions: Grow rapidly and are characterized by their base raised to a variable exponent.

Examples

• A relation defined as y = 2x + 3 is a function.
• A relation like x² + y² = r² represents a circle and is not a function.

Conclusion

• Understanding the distinction between functions and non-functions is critical for working with graphs in mathematics, impacting problem-solving and analysis techniques used across various mathematical concepts.

Test your knowledge on the differences between functions and non-functions through these flashcards. Each card presents a key term and its definition, helping you understand the fundamental concepts of graphing and relationships in mathematics.