Piecewise-Defined Functions: Concepts, Evaluation, Graphs, and Applications

Piecewise-Defined Functions

Definition and Structure

A piecewise-defined function is a function composed of multiple equations, each applying to a specific interval or condition within the domain. The function's rule specifies which equation to use for each part of the domain.

• Key Point 1: The function is defined by different expressions depending on the value of the independent variable (usually x).

• Key Point 2: The rule includes both the equations and the conditions (such as inequalities or equalities) that determine which equation applies.

• Example: $f(x) = \begin{cases} x^2 & \text{if } x < 2 \\ -3x + 12 & \text{if } x \geq 2 \end{cases}$

Evaluating Piecewise-Defined Functions

To evaluate a piecewise-defined function at a specific value, determine which condition the value satisfies and use the corresponding equation.

• Key Point 1: Check the value of x against the conditions for each piece.

• Key Point 2: Substitute the value into the appropriate equation.

• Example: For $f(x) = \begin{cases} x^2 & x < 2 \\ -3x + 12 & x \geq 2 \end{cases}$:

  • $f(0) = 0^2 = 0$ (since $0 < 2$)

  • $f(2) = -3 \times 2 + 12 = -6 + 12 = 6$ (since $2 \geq 2$)

  • $f(4) = -3 \times 4 + 12 = -12 + 12 = 0$ (since $4 \geq 2$)

Finding Intercepts of Piecewise-Defined Functions

Intercepts are points where the function crosses the axes. For piecewise-defined functions, check each piece for possible intercepts.

• Key Point 1: x-intercepts: Set $f(x) = 0$ and solve for x in each piece.

• Key Point 2: y-intercept: Evaluate $f(0)$ using the appropriate piece.

• Example: For $f(x) = \begin{cases} x^2 & x < 2 \\ -3x + 12 & x \geq 2 \end{cases}$, $f(0) = 0$ (y-intercept at (0,0)).

Graphing Piecewise-Defined Functions

Graphing a piecewise-defined function involves plotting each piece over its specified interval and marking any discontinuities or endpoints.

• Key Point 1: Plot each equation only over its valid domain.

• Key Point 2: Use open or closed circles to indicate whether endpoints are included or excluded.

• Example: The graph of $f(x) = \begin{cases} x^2 & x < 2 \\ -3x + 12 & x \geq 2 \end{cases}$ will show a parabola for $x < 2$ and a straight line for $x \geq 2$, with appropriate endpoint markings.

Using Graphs to Determine Rules for Piecewise-Defined Functions

Sometimes, the rule for a piecewise-defined function can be deduced from its graph by identifying the equations and the intervals where they apply.

• Key Point 1: Analyze the graph to see where the function changes its behavior.

• Key Point 2: Match each segment to a known function (e.g., linear, quadratic).

• Example: A graph with a line for $x < -3$ and a constant value for $x = -3$ suggests a rule like $f(x) = \begin{cases} -x + 3 & x \neq -3 \\ 1 & x = -3 \end{cases}$.

Applications of Piecewise-Defined Functions

Piecewise-defined functions are often used to model real-world situations where rules change based on conditions, such as tax brackets or shipping costs.

• Key Point 1: Identify the different cases in the application (e.g., income ranges for tax rates).

• Key Point 2: Write a function that describes the output for each case.

• Example: On the planet Sarnun, the tax function is: $T(x) = \begin{cases} 0.05x & \text{if } x \leq 28,000 \\ 0.05 \times 28,000 + 0.07(x - 28,000) & \text{if } x > 28,000 \end{cases}$

  • a. For $x = 15,000$, $T(15,000) = 0.05 \times 15,000 = 750$ dollars.

  • b. For $x = 30,000$, $T(30,000) = 0.05 \times 28,000 + 0.07 \times (2,000) = 1,400 + 140 = 1,540$ dollars.

  • c. The function above describes the taxes owed as a function of income.

Summary Table: Properties of Piecewise-Defined Functions

Property	Description
Definition	Uses multiple equations for different intervals
Evaluation	Choose the correct equation based on the input value
Graphing	Plot each piece over its domain; mark endpoints
Applications	Model real-world scenarios with changing rules