Piecewise Function Modeling: Functions and Their Modeling - Mathematics (Grade 11)

When to Use Piecewise Functions

A piecewise function is useful when one situation cannot be described well by one formula. The domain is split into intervals, and each interval uses its own rule.

For example, electricity cost can be cheaper for the first usage tier and more expensive after a certain boundary. A boundary like this is a sign that the model should be written in pieces.

Mathematical Definition

A piecewise function can be written in the form:

f(x) = \begin{cases} f_1(x), & \text{if } x \in D_1 \\ f_2(x), & \text{if } x \in D_2 \\ f_3(x), & \text{if } x \in D_3 \\ \vdots \\ f_n(x), & \text{if } x \in D_n \end{cases}

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

When reading or building a piecewise function, pay attention to three things:

The formula on each row.
The interval condition that decides when that formula is used.
The function value at connection points, especially when an interval boundary uses $\lt$ , $\leq$ , $\gt$ , or $\geq$ .

Common Piece Shapes

Linear Pieces

A linear piecewise function uses a linear function on each interval. This form often appears when the rate of change is constant for a while, then changes after a certain point.

Linear Piecewise Function

Example of a linear piecewise function with three different pieces.

The function above can be written as:

f(x) = \begin{cases} 2x + 3, & \text{if } -2 \leq x < 0 \\ -x + 3, & \text{if } 0 \leq x < 2 \\ 0.5x, & \text{if } x \geq 2 \end{cases}

The three rows are not three separate functions. They work together as one function, and the chosen formula depends on the value of $x$ .

Quadratic Pieces

Piecewise functions can also use quadratic pieces. This is helpful when one part of the model curves instead of following a straight line.

Combined Piecewise Function

Combination of quadratic and linear functions in one piecewise function.

Connection Points and Continuity

Connections Without Jumps

A piecewise function is continuous if the graph does not jump at a connection point. The value from the left, the value from the right, and the function value at that point must match.

Condition for continuity at point $x = c$ :

\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)

Example of a continuous piecewise function:

f(x) = \begin{cases} x + 1, & \text{if } x < 2 \\ 3, & \text{if } x = 2 \\ 5 - x, & \text{if } x > 2 \end{cases}

For continuity at $x = 2$ :

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

Connections With Jumps

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

Discontinuous Piecewise Function

Example of a function with jump discontinuity at

x = 1

Modeling with Piecewise Functions

Many real-world situations can be modeled with piecewise functions because their rules change after certain boundaries. Progressive taxes, tiered parking fees, and tiered electricity rates are common examples.

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:

C(x) = \begin{cases} 1000x, & \text{if } 0 \leq x \leq 50 \\ 50000 + 1500(x-50), & \text{if } 50 < x \leq 100 \\ 125000 + 2000(x-100), & \text{if } x > 100 \end{cases}

The number $50{,}000$ in the second row comes from the first $50 \text{ kWh}$ . The number $125{,}000$ in the third row comes from the cost up to the first $100 \text{ kWh}$ . This keeps the cost from restarting at $0$ when the usage enters a new interval.

Electricity cost table:

Usage (kWh)	$30$	$50$	$75$	$100$	$150$
Cost (Rp)	$30{,}000$	$50{,}000$	$87{,}500$	$125{,}000$	$225{,}000$

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:

s(t) = \begin{cases} 5t, & \text{if } 0 \leq t \leq 0.5 \\ 2.5 + 15(t-0.5), & \text{if } 0.5 < t \leq 1.5 \\ 17.5 + 60(t-1.5), & \text{if } 1.5 < t \leq 2 \end{cases}

The constants $2.5$ and $17.5$ keep the total distance moving forward from the last position, instead of returning to $0$ when the transportation mode changes.

Determining Piecewise Function Equations

To determine piecewise function equations from graphs or situations:

Split the domain at every boundary where the rule changes.
Determine the formula for each interval.
Check the connection points so the interval signs and function values do not conflict.
Write all pieces 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 $x = 4$

The solution can be summarized like this:

Interval	Graph information	Equation
$[-2, 0)$	Passes through $(-2, 0)$ with slope $2$	$y = 2(x + 2) = 2x + 4$
$[0, 2)$	Horizontal line	$y = 4$
$[2, 4]$	Passes through $(2, 4)$ with slope $-1$	$y = -1(x - 2) + 4 = -x + 6$

Piecewise function:

f(x) = \begin{cases} 2x + 4, & \text{if } -2 \leq x < 0 \\ 4, & \text{if } 0 \leq x < 2 \\ -x + 6, & \text{if } 2 \leq x \leq 4 \end{cases}

Practice Problems

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

$f(x) = \begin{cases} x^2, & \text{if } x < 2 \\ 2x + 1, & \text{if } 2 \leq x < 4 \\ 9, & \text{if } x \geq 4 \end{cases}$
An online taxi company applies the following rates:

The base fare is $\text{Rp }10{,}000$ for the first $2 \text{ km}$ . Km $3\text{-}10$ costs $\text{Rp }4{,}000/\text{km}$ . Above $10 \text{ km}$ , the additional rate becomes $\text{Rp }3{,}000/\text{km}$ .

Create a piecewise function model for the total cost!
Determine whether the following function is continuous at $x = 1$ :

$g(x) = \begin{cases} 3x - 1, & \text{if } x < 1 \\ 2, & \text{if } x = 1 \\ x + 1, & \text{if } x > 1 \end{cases}$
Sketch the graph of the function:

$h(x) = \begin{cases} -x + 2, & \text{if } x < 0 \\ x^2, & \text{if } 0 \leq x < 2 \\ 4, & \text{if } x \geq 2 \end{cases}$
A worker is paid with the following system:

The first $8 \text{ hours}$ are paid at $\text{Rp }50{,}000/\text{hour}$ . Overtime from the $9$ th hour onward is paid at $\text{Rp }75{,}000/\text{hour}$ .

If the maximum work is $12 \text{ hours/day}$ , create a daily wage function!

Answer Key

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 $f(x) = 9$

$f(5) = 9$
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:
$C(x) = \begin{cases} 10000, & \text{if } 0 < x \leq 2 \\ 4000x + 2000, & \text{if } 2 < x \leq 10 \\ 3000x + 12000, & \text{if } x > 10 \end{cases}$
Checking continuity:

At $x = 1$ :
$\lim_{x \to 1^-} g(x) = \lim_{x \to 1^-} (3x - 1) = 3(1) - 1 = 2$
Since $\lim_{x \to 1^-} g(x) = \lim_{x \to 1^+} g(x) = g(1) = 2$ , the function is continuous at $x = 1$ .
Sketch of graph $h(x)$ :

Graph of Function $h(x)$
Piecewise function with three parts: decreasing linear, quadratic, and constant
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:
$U(t) = \begin{cases} 50000t, & \text{if } 0 < t \leq 8 \\ 75000t - 200000, & \text{if } 8 < t \leq 12 \end{cases}$