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:

• $0\text{-}50 \text{ kWh}$: $\text{Rp }1{,}000/\text{kWh}$
• $51\text{-}100 \text{ kWh}$: $\text{Rp }1{,}500/\text{kWh}$
• $>100 \text{ kWh}$: $\text{Rp }2{,}000/\text{kWh}$

The mathematical model:

Electricity cost table:

Staged Speed

Example: Multi-Modal Journey

Someone takes a journey with:

• Walking: $5 \text{ km/h}$ for $0.5 \text{ hours}$
• Cycling: $15 \text{ km/h}$ for $1 \text{ hour}$
• Driving: $60 \text{ km/h}$ for $0.5 \text{ hours}$

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: $\text{Rp }10{,}000$ (for the first $2 \text{ km}$)

• Km $3\text{-}10$: $\text{Rp }4{,}000/\text{km}$

  • Above $10 \text{ km}$: $\text{Rp }3{,}000/\text{km}$

  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 $8 \text{ hours}$: $\text{Rp }50{,}000/\text{hour}$
   • Overtime ($9$th hour onwards): $\text{Rp }75{,}000/\text{hour}$

   If the maximum work is $12 \text{ hours/day}$, 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:

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