Discrete/piecewise models - Mathematics: Applications & Interpretation IB Study Notes
Overview
# Discrete/Piecewise Models Summary ## Key Learning Outcomes Students learn to construct and analyze discrete functions and piecewise-defined models that represent real-world scenarios with distinct domains or different rules across intervals. This includes understanding step functions (such as postage rates or parking fees), modeling taxation systems with progressive rates, and interpreting graphs where function behavior changes at specific boundaries. Essential skills include determining function values at transition points, sketching accurate piecewise graphs with appropriate notation (open/closed circles), and writing piecewise function rules from contextual descriptions. ## Exam Relevance This topic frequently appears in Paper 1 and Paper 2, typically worth 6-9 marks per question, often integrated with optimization, financial mathematics, or rate-of-change problems. Students must demonstrate proficiency in determining domains of applicability, calculating outputs across different intervals, and using GDC technology to verify solutions while showing clear analytical working
Core Concepts & Theory
Discrete functions map inputs to outputs where the domain consists of separate, distinct values (often integers), rather than continuous intervals. Think of them as functions that "jump" from point to point—you can count the possible input values.
Piecewise functions (also called hybrid functions) are defined by different expressions over different intervals of their domain. The general form is:
$$f(x) = \begin{cases} f_1(x) & \text{if } x \in D_1 \ f_2(x) & \text{if } x \in D_2 \ \vdots & \vdots \ f_n(x) & \text{if } x \in D_n \end{cases}$$
where $D_1, D_2, \ldots, D_n$ are non-overlapping subsets of the domain.
Key terminology:
- Domain restrictions: Boundaries where the function rule changes (critical for continuity)
- Step functions: Special piecewise functions that are constant over each interval (e.g., postal rates, parking fees)
- Continuous vs. discontinuous: A piecewise function is continuous at $x = a$ if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$
Essential notation: Use square brackets [ ] for inclusive boundaries and parentheses ( ) for exclusive ones. For example, $[2, 5)$ means $2 \leq x < 5$.
Cambridge tip: Always verify that your piecewise definition covers the entire relevant domain with no gaps or overlaps. Exam questions often test boundary behavior specifically.
Mnemonic for continuity: "Left meets right at the height" — left-hand limit equals right-hand limit equals function value.
Detailed Explanation with Real-World Examples
Taxi fare modeling provides an excellent real-world context. A typical fare structure:
$$C(d) = \begin{cases} 5 & \text{if } 0 \leq d \leq 2 \ 5 + 2(d-2) & \text{if } d > 2 \end{cases}$$
where $C$ is cost in dollars and $d$ is distance in kilometers. The flat fee ($5) applies for the first 2 km, then $2/km thereafter. This creates a continuous piecewise function—no sudden jumps in cost.
Postal rates demonstrate discontinuous step functions:
$$P(m) = \begin{cases} 0.85 & \text{if } 0 < m \leq 20 \ 1.20 & \text{if } 20 < m \leq 50 \ 1.75 & \text{if } 50 < m \leq 100 \end{cases}$$
where mass $m$ is in grams. Notice the discontinuities—a 21g letter costs $1.20, not marginally more than $0.85.
Income tax brackets are classic piecewise linear models:
$$T(I) = \begin{cases} 0 & \text{if } I \leq 12570 \ 0.20(I - 12570) & \text{if } 12570 < I \leq 50270 \ 7540 + 0.40(I - 50270) & \text{if } I > 50270 \end{cases}$$
Analogy: Think of piecewise functions as a journey with different speed limits in different zones. Your speedometer reading (function value) changes based on which zone (interval) you're in.
Discrete functions model scenarios like:
- Number of cinema tickets ($n$ people → cost)
- Mobile data plans (per GB pricing)
- Inventory systems (whole units only)
Real-world connection: Most real-world pricing, taxation, and resource allocation involves piecewise modeling due to tiered structures and practical constraints.
Worked Examples & Step-by-Step Solutions
**Example 1:** A parking garage charges $\$4$ for the first hour, then $\$2.50$ for each additional hour or part thereof, up to 8 hours maximum. *(a)* Write a piecewise function $C(t)$ for $0 < t \leq 8$ **Solution:** $$C(t) = \begin{cases} 4 & \text{if } 0 < t \leq 1 \\ 4 + 2.5\lceil t-1 \rceil &...
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Key Concepts
- Discrete Model: A mathematical description where values can only be distinct, separate numbers, often whole numbers, like counting individual items.
- Piecewise Function: A function defined by multiple sub-functions, each applied to a different interval of the input variable.
- Interval: A set of numbers between two specified values, often used to define the 'pieces' of a piecewise function.
- Sub-function: One of the individual rules or equations that make up a piecewise function, each valid for a specific interval.
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Exam Tips
- →When asked to define a piecewise function, always include both the equation for each piece AND the interval (conditions) for which it applies.
- →For graphing, use open and closed circles correctly at the boundary points to show whether the point is included or excluded.
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