Study Guide: Piecewise Functions (Calculus Prerequisites)

Piecewise Functions

Introduction to Piecewise Functions

Piecewise functions are an important concept in calculus prerequisites, as they allow us to define a function using different expressions depending on the input value. This is especially useful for modeling situations where a rule changes at certain points, and is foundational for understanding limits and continuity in calculus.

• Definition: A piecewise function is a single function composed of two or more equations, each valid for a specific interval of the domain.

• Notation: Piecewise functions are typically written using curly braces to indicate the different cases.

• Continuity: If the y-values (outputs) at the boundaries between pieces do not match, the function has a jump discontinuity at those points.

• Graphing: When graphing, use open or closed circles to indicate whether endpoints are included or excluded.

Example:

• Consider the function: $f(x) = \begin{cases} -x & \text{if } x < -1 \\ x^2 - 4 & \text{if } x \geq -1 \end{cases}$

• To evaluate $f(-3)$, $f(-1)$, and $f(2)$, substitute each value into the appropriate equation:

  • $f(-3) = -(-3) = 3$ (since $-3 < -1$)

  • $f(-1) = (-1)^2 - 4 = 1 - 4 = -3$ (since $-1 \geq -1$)

  • $f(2) = (2)^2 - 4 = 4 - 4 = 0$ (since $2 \geq -1$)

Evaluating Piecewise Functions

To evaluate a piecewise function at a given input, determine which interval the input falls into and use the corresponding expression.

• Step 1: Identify the interval for the input value.

• Step 2: Substitute the input into the correct formula.

• Step 3: Compute the result.

Practice Example:

• Given: $f(x) = \begin{cases} 4 & \text{if } x < 3 \\ x - 3 & \text{if } 3 \leq x < 8 \\ x & \text{if } x \geq 8 \end{cases}$

• Evaluate $f(-1)$:

  • Since $-1 < 3$, use $f(x) = 4$.

  • $f(-1) = 4$

• Evaluate $f(5)$:

  • Since $3 \leq 5 < 8$, use $f(x) = x - 3$.

  • $f(5) = 5 - 3 = 2$

Graphing Piecewise Functions

Graphing a piecewise function involves plotting each segment according to its formula and domain. Pay attention to endpoints, using open circles for excluded values and closed circles for included values.

• Example: $f(x) = \begin{cases} -4 & \text{if } x < -2 \\ x + 1 & \text{if } -2 \leq x < 1 \\ x & \text{if } x \geq 1 \end{cases}$

• For $x < -2$, plot a horizontal line at $y = -4$.

• For $-2 \leq x < 1$, plot the line $y = x + 1$ from $x = -2$ to $x = 1$.

• For $x \geq 1$, plot the line $y = x$ for $ x \geq 1 $.

• At $x = -2$ and $x = 1$, check which expressions include the endpoint and use closed or open circles accordingly.

Summary Table: Piecewise Function Structure

Interval	Expression	Endpoint Inclusion
$x < a$	First formula	Open circle at $x = a$
$a \leq x < b$	Second formula	Closed circle at $x = a$, open at $x = b$
$x \geq b$	Third formula	Closed circle at $x = b$

Applications: Piecewise functions are used to model tax brackets, shipping costs, and physical phenomena with abrupt changes.

Additional info: Piecewise functions are foundational for understanding limits, continuity, and later topics in calculus such as integration and differentiation of non-continuous functions.