Pre-Calculus: Piecewise-Defined Functions

Piecewise-defined functions are the mathematical equivalent of a multi-tool, allowing you to model complex, real-world behavior that can't be described by a single, simple rule. They are fundamental in engineering for describing systems that operate differently under varying conditions, such as tax brackets, shipping rates, or the behavior of electrical circuits. Mastering them involves learning to read, evaluate, graph, and construct these versatile functions, with a special focus on what happens at the boundaries where the rules change.

What Is a Piecewise-Defined Function?

A piecewise-defined function is a function that is defined by two or more formulas, each applying to a different part of its domain (the set of all possible input values). Instead of one rule for every input $x$, you have a set of rules and a set of intervals that tell you which rule to use. The function is "pieced together" from these different components.

The notation for a piecewise function uses a large left brace to group the function rules and their associated domains. A classic example is the absolute value function, which can be written piecewise as:

$$f(x)=|x|=\{\begin{array}{ll}x,& \text{if }x\ge 0\\ -x,& \text{if }x<0\end{array}$$

This reads: "For $x$ values greater than or equal to 0, use the rule $f(x)=x$. For $x$ values less than 0, use the rule $f(x)=-x$." The domain of this function is all real numbers, but the rule used to find the output changes at $x=0$.

Evaluating Piecewise Functions

Evaluating a piecewise function is a straightforward, two-step process: First, determine which interval or "piece" of the domain contains your input value. Second, use the corresponding formula to compute the output. Accuracy depends entirely on careful attention to the inequality conditions.

Let's evaluate the function $g(x)$ at specific points.

$$g(x)=\{\begin{array}{ll}{x}^2+1,& \text{if }x\le 2\\ 5,& \text{if }2<x<4\\ 2x-3,& \text{if }x\ge 4\end{array}$$

• To find $g(1)$: The input is $x=1$. Which condition does it satisfy? $1\le 2$ is true, so we use the first piece: $g(1)=(1{)}^2+1=2$.
• To find $g(3)$: The input is $x=3$. Check the conditions: $3\le 2$ is false. $2<3<4$ is true, so we use the second piece: $g(3)=5$.
• To find $g(4)$: The input is $x=4$. $4\le 2$ is false. $2<4<4$ is false (4 is not less than 4). $4\ge 4$ is true, so we use the third piece: $g(4)=2(4)-3=5$.

Notice that at $x=2$, the function uses the first piece ($g(2)=2^2+1=5$). At $x=4$, it uses the third piece. The constant second piece only applies for inputs strictly between 2 and 4.

Graphing Piecewise Functions

Graphing is where you visually "see" the pieces come together. You graph each sub-function, but only for the portion of its domain specified in the piecewise definition.

Process:

1. Identify each piece and its domain interval.
2. Graph the equation for that piece as if it were defined forever.
3. Erase the parts of the graph that lie outside the specified domain interval.
4. Pay critical attention to the endpoints. Use a closed dot (•) to indicate an endpoint that is included in the piece (using $\le$ or $\ge$). Use an open dot (◦) to indicate an endpoint that is not included (using $<$ or $>$).

Let's graph a simple function:

$$h(x)=\{\begin{array}{ll}-2x,& \text{if }x<1\\ x+1,& \text{if }x\ge 1\end{array}$$

• First piece ($-2x$, for $x<1$): This is a line with slope -2. We graph it for $x$-values less than 1. At the boundary $x=1$, we would calculate $h(1)=-2(1)=-2$, but this rule does not apply at $x=1$. So, we plot an open dot at the point $(1,-2)$.
• Second piece ($x+1$, for $x\ge 1$): This is a line with slope 1. We graph it for $x$-values greater than or equal to 1. At the boundary $x=1$, we calculate $h(1)=1+1=2$ and plot a closed dot at $(1,2)$.

The final graph shows a line coming from the left ending at an open dot at $(1,-2)$, and a different line starting at a closed dot at $(1,2)$ and going to the right. This visual jump is a key feature.

Determining Continuity at Boundary Points

A function is continuous at a point if you can draw it at that point without lifting your pencil. Formally, three conditions must hold: 1) The function is defined at the point. 2) The limit exists at the point. 3) The limit equals the function value. For piecewise functions, boundaries are where continuity is most likely to break.

Continuity Check at a Boundary $x=a$:

1. Find the left-hand limit (${\lim }_{x\to a^{-}}f(x)$): Use the rule defined for $x<a$.
2. Find the right-hand limit (${\lim }_{x\to a^{+}}f(x)$): Use the rule defined for $x>a$.
3. Find the function value $f(a)$: Use the rule that includes $x=a$.
4. The function is continuous at $x=a$ if: ${\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a^{+}}f(x)=f(a)$.

Consider our function $h(x)$ from the graphing example at $x=1$.

• Left-hand limit: Use the first piece. ${\lim }_{x\to 1^{-}}h(x)={\lim }_{x\to 1^{-}}(-2x)=-2$.
• Right-hand limit: Use the second piece. ${\lim }_{x\to 1^{+}}h(x)={\lim }_{x\to 1^{+}}(x+1)=2$.
• Function value: $h(1)=2$ from the second piece.

Since $-2\ne 2$, the limits are not equal, so the overall limit does not exist. Therefore, $h(x)$ is not continuous at $x=1$. The graph shows this as a jump.

Writing Piecewise Functions for Real-World Scenarios

This is the ultimate application skill. You translate a written description of a scenario into a precise mathematical model.

Example (Shipping Costs): A company charges a flat rate of \$5.00 for shipping any order weighing up to 1 lb. For orders over 1 lb but up to 5 lbs, they charge \$8.00. Orders over 5 lbs are charged \$8.00 plus \$1.50 for each additional pound over 5.

Step 1: Identify the variable. Let $w$ = weight of the package in pounds. Let $C(w)$ = shipping cost in dollars. Step 2: Identify the different "pieces" of the domain. The rules change at $w=1$ lb and $w=5$ lbs. Step 3: Define the rule for each piece.

• Piece 1 ($0<w\le 1$): A constant \$5.00. So, $C(w)=5$.
• Piece 2 ($1<w\le 5$): A constant \$8.00. So, $C(w)=8$.
• Piece 3 ($w>5$): \$8.00 plus \$1.50 per pound over 5. The pounds over 5 is $(w-5)$. So, $C(w)=8+1.50(w-5)$, which simplifies to $C(w)=1.5w+0.5$.

Step 4: Assemble the function.

$$C(w)=\{\begin{array}{ll}5,& \text{if }0<w\le 1\\ 8,& \text{if }1<w\le 5\\ 1.5w+0.5,& \text{if }w>5\end{array}$$

This function now allows you to calculate the exact cost for any package weight.

Common Pitfalls

1. Misreading the Domain for Evaluation: The most frequent error is using the wrong formula because you misapply the inequality at a boundary point. Correction: Always substitute the input value into the inequality conditions first, not the formulas. Check each condition systematically from top to bottom until you find the one that is true.
2. Incorrect Endpoint Plotting on Graphs: Using an open dot when a closed dot is needed, or vice versa, misrepresents the function's definition. Correction: Associate the type of inequality directly with the dot. $\le$ or $\ge$ means the point is part of that piece's graph → use a closed dot. $<$ or $>$ means the point is not part of that piece's graph → use an open dot.
3. Assuming Continuity or Discontinuity: Assuming a function is continuous because the formulas meet, or discontinuous because they are different. Correction: You must test continuity using the three-part limit definition at every boundary point. The formulas could meet at the exact point (making it continuous), or the same formula could be defined on both sides but with a hole on one side (making it discontinuous).
4. Domain Errors in Real-World Models: Forgetting to state the practical domain or using "$\le$" when you should use "$<$". Correction: In contexts like shipping, note if inequalities are strict (e.g., weight must be $>0$). Pay close attention to phrases like "over 5 lbs" ($>5$) versus "5 lbs or more" ($\ge 5$).

Summary

• A piecewise-defined function uses multiple sub-functions, each with its own specified interval of the overall domain, to define a single rule of correspondence.
• To evaluate, first identify which domain interval contains your input value, then apply the corresponding formula.
• To graph, plot each sub-function individually but only over its specified interval, using open or closed dots to correctly indicate inclusion or exclusion of endpoints.
• Continuity at a boundary requires the left-hand limit, right-hand limit, and function value to all be equal. Piecewise functions often have points of discontinuity (jumps) at these boundaries.
• The power of piecewise functions lies in modeling real-world scenarios where a process follows different rules under different conditions, such as tiered pricing, tax codes, or signal processing in engineering.