Piecewise-Defined Functions


Find a formula for each function graphed in Exercises 29–32.


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Verified step by step guidance

1. Identify the different segments of the piecewise function from the graph. Look for changes in slope, discontinuities, or different behaviors in different intervals.
2. Determine the domain for each segment. This involves identifying the x-values over which each segment is defined.
3. For each segment, determine the type of function it represents (e.g., linear, quadratic, constant) by analyzing the graph's shape and behavior.
4. Write the equation for each segment using the appropriate function type. For linear segments, use the slope-intercept form: $y=\mathrm{mx}+b$, where m is the slope and b is the y-intercept.
5. Combine the equations into a piecewise function, specifying the domain for each segment. Use the format: $f\left(x\right)=\{\begin{array}{cc}\mathrm{expression}& ,\mathrm{condition}\end{array}$.

Key Concepts

Piecewise-Defined Functions

Piecewise-defined functions are functions that have different expressions or rules for different intervals of the domain. They are useful for modeling situations where a function behaves differently in different scenarios. Understanding how to interpret and construct these functions is crucial for analyzing graphs that change behavior at specific points.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of a function to understand its behavior, such as identifying intervals, slopes, and points of discontinuity. This skill is essential for translating a graph into a piecewise-defined function, as it helps determine the different expressions that apply to each segment of the graph.

Function Formulation

Function formulation is the process of creating mathematical expressions that accurately represent a function's behavior. For piecewise-defined functions, this involves writing separate expressions for each interval of the domain, ensuring continuity and correct representation of the graph's features. Mastery of this concept allows for precise modeling of complex functions.