Lecture Notes on Functions
Recap of Previous Lecture
- Relation is a function if all x-values are different.
- Use the vertical line test: if a vertical line crosses more than one point, it is not a function.
- One-to-One Functions:
- Definition: If each x-value maps to one unique y-value.
- Horizontal line test: The function is one-to-one if a horizontal line touches the graph at only one point.
- Onto Functions:
- Definition: The range is equal to the co-domain.
- Often, the co-domain is assumed to be all real numbers.
Today's Topics
Discrete vs Continuous Functions
- Discrete Functions:
- Represented by individual points (e.g., scatter plots).
- Real-world situations where only certain numbers are reasonable.
- Continuous Functions:
- Graph is a line or unbroken curve.
- All real numbers are reasonable.
- Discontinuous Functions:
- Mix of connected and unconnected points.
- Discrete functions are a type of discontinuous function.
Intervals
- Definition: Set of all real numbers between two given numbers.
- Examples:
- Interval notation:
(-2 < x < 5)
- Infinite intervals:
(y ≥ 1)
Determining Continuity
- Continuous Functions:
- Unbroken curve, no holes.
- Domain and range cover all real numbers.
- Neither Continuous nor Discrete:
- Has breaks or interruptions.
- Discrete Functions:
- Individual points, specific x-values.
Coffee Bean Pricing Example
- Mixed continuous (up to 2 pounds) and discrete (specific weights) function.
- Visually visualizing helps in understanding the nature of the function.
Interval and Set Builder Notation
- Set Builder Notation:
- Uses braces
{} to indicate a set.
- Vertical line
| means "such that."
- Symbol
∈ means "element of."
- Interval Notation:
- Uses parentheses
() for non-inclusive, brackets [] for inclusive.
(-∞, 2) means values less than 2.
Examples:
- All Real Numbers:
- Set Builder:
{x | x ∈ ℝ}
- Interval:
(-∞, ∞)
Practice and Examples
- Practice writing domain and range in both set builder and interval notation.
- Recognize how discontinuous functions might require combinations of intervals using the union symbol (∪).
Upcoming Tasks
- Homework: Questions 1-39 (odds only). Due tomorrow.
- Be prepared to discuss and review answers in the next class.
Use these notes to solidify your understanding of functions, particularly the distinctions between one-to-one, onto, discrete, continuous, and discontinuous functions. Practice using set builder and interval notation for both domain and range.