Piecewise Function Modeling

Understanding Piecewise Functions

A piecewise function is a function that is defined by several different equations on certain intervals of its domain. Each "piece" of the function applies to a specific part of the domain.

Mathematical Definition

A piecewise function can be written in the form:

where $D_1, D_2, D_3, \ldots, D_n$ are intervals that form a partition of the function's domain.

Characteristics of piecewise functions:

• Consists of several different equations
• Each equation applies to a specific interval
• Can be continuous or discontinuous
• Intervals do not overlap

Types of Piecewise Functions

Linear Piecewise Functions

A linear piecewise function is a function where each piece is a linear function.

The function above can be written as:

Quadratic Piecewise Functions

Piecewise functions can also contain quadratic pieces or combinations of linear and quadratic functions.

Continuity of Piecewise Functions

Continuous Piecewise Functions

A piecewise function is said to be continuous if there are no "jumps" at the connection points between pieces.

Condition for continuity at point $x = c$:

Example of a continuous piecewise function:

For continuity at $x = 2$:

• $\lim_{x \to 2^-} (x + 1) = 3$
• $\lim_{x \to 2^+} (5 - x) = 3$
• $f(2) = 3$

Discontinuous Piecewise Functions

Discontinuous piecewise functions have "jumps" or "holes" at certain points.

Modeling with Piecewise Functions

Progressive Rates

Many real-world situations can be modeled with piecewise functions, such as progressive tax rates or tiered parking fees.

Example: Electricity Rates

An electricity company applies tiered rates:

• 0-50 kWh: Rp 1,000/kWh
• 51-100 kWh: Rp 1,500/kWh

• 100 kWh: Rp 2,000/kWh

The mathematical model:

Electricity cost table:

Staged Speed

Example: Multi-Modal Journey

Someone takes a journey with:

• Walking: 5 km/h for 0.5 hours
• Cycling: 15 km/h for 1 hour
• Driving: 60 km/h for 0.5 hours

Distance function with respect to time:

Determining Piecewise Function Equations

To determine piecewise function equations from graphs or situations:

1. Identify the intervals of the domain
2. Determine the equation for each interval
3. Check continuity at connection points
4. Write in piecewise notation

Example:

From a graph showing:

• Line with slope 2 from x = -2 to x = 0
• Horizontal line y = 4 from x = 0 to x = 2
• Line with slope -1 from x = 2 to x = 4

Solution steps:

1. Interval 1: $[-2, 0)$

   • Passes through (-2, 0) with slope 2
   • Equation: $y = 2(x + 2) = 2x + 4$

2. Interval 2: $[0, 2)$

   • Horizontal line
   • Equation: $y = 4$

3. Interval 3: $[2, 4]$

   • Passes through (2, 4) with slope -1
   • Equation: $y = -1(x - 2) + 4 = -x + 6$

Piecewise function:

Practice Problems

1. Determine the values of $f(1)$, $f(3)$, and $f(5)$ for the function:

   $$f(x) = \begin{cases} x^2, & \text{if } x < 2 \\ 2x + 1, & \text{if } 2 \leq x < 4 \\ 9, & \text{if } x \geq 4 \end{cases}$$

2. An online taxi company applies the following rates:

   • Base fare: Rp 10,000 (for the first 2 km)
   • Km 3-10: Rp 4,000/km
   • Above 10 km: Rp 3,000/km

   Create a piecewise function model for the total cost!

3. Determine whether the following function is continuous at $x = 1$:

   $$g(x) = \begin{cases} 3x - 1, & \text{if } x < 1 \\ 2, & \text{if } x = 1 \\ x + 1, & \text{if } x > 1 \end{cases}$$

4. Sketch the graph of the function:

   $$h(x) = \begin{cases} -x + 2, & \text{if } x < 0 \\ x^2, & \text{if } 0 \leq x < 2 \\ 4, & \text{if } x \geq 2 \end{cases}$$

5. A worker is paid with the following system:

   • First 8 hours: Rp 50,000/hour
   • Overtime (9th hour onwards): Rp 75,000/hour

   If the maximum work is 12 hours/day, create a daily wage function!

Answer Key

1. Calculating function values:

   For $f(1)$: since $1 < 2$, use $f(x) = x^2$

   $$f(1) = 1^2 = 1$$

   For $f(3)$: since $2 \leq 3 < 4$, use $f(x) = 2x + 1$

   $$f(3) = 2(3) + 1 = 7$$

   For $f(5)$: since $5 \geq 4$, use $f(x) = 9$

   $$f(5) = 9$$

2. Taxi fare model:

   Let $x$ be the distance in km, then:

   $$C(x) = \begin{cases} 10000, & \text{if } 0 < x \leq 2 \\ 10000 + 4000(x-2), & \text{if } 2 < x \leq 10 \\ 10000 + 32000 + 3000(x-10), & \text{if } x > 10 \end{cases}$$

   Or simplified:

   $$C(x) = \begin{cases} 10000, & \text{if } 0 < x \leq 2 \\ 4000x + 2000, & \text{if } 2 < x \leq 10 \\ 3000x + 12000, & \text{if } x > 10 \end{cases}$$

3. Checking continuity:

   At $x = 1$:

   $$\lim_{x \to 1^-} g(x) = \lim_{x \to 1^-} (3x - 1) = 3(1) - 1 = 2$$

   $$\lim_{x \to 1^+} g(x) = \lim_{x \to 1^+} (x + 1) = 1 + 1 = 2$$

   $$g(1) = 2$$

   Since $\lim_{x \to 1^-} g(x) = \lim_{x \to 1^+} g(x) = g(1) = 2$, the function is continuous at $x = 1$.

4. Sketch of graph $h(x)$:

   Graph of Function $h(x)$

   Piecewise function with three parts: decreasing linear, quadratic, and constant

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5. Daily wage function:

   Let $t$ be the working hours, then:

   $$U(t) = \begin{cases} 50000t, & \text{if } 0 < t \leq 8 \\ 400000 + 75000(t-8), & \text{if } 8 < t \leq 12 \end{cases}$$

   Or simplified:

   $$U(t) = \begin{cases} 50000t, & \text{if } 0 < t \leq 8 \\ 75000t - 200000, & \text{if } 8 < t \leq 12 \end{cases}$$