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 intervals on the x-axis where the function changes its behavior. This will help in determining the different pieces of the piecewise function.
2. For each interval, observe the type of function that is graphed. Common types include linear, quadratic, or constant functions.
3. Determine the equation for each piece of the function. For linear pieces, use the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For quadratic pieces, use the standard form \( y = ax^2 + bx + c \).
4. Ensure that the function is continuous at the points where the pieces meet, unless the graph indicates a jump discontinuity. Adjust the equations if necessary to match the graph.
5. Combine the equations into a single piecewise function, specifying the domain for each piece. Use the format: \( f(x) = \begin{cases} \text{equation 1,} & \text{if } x \text{ is in interval 1} \\ \text{equation 2,} & \text{if } x \text{ is in interval 2} \\ \end{cases} \).

Key Concepts

Piecewise-Defined Functions

Piecewise-defined functions are functions that have different expressions or formulas for different intervals of the input variable. They are useful for modeling situations where a rule changes based on the input value, such as tax brackets or speed limits. Understanding how to interpret and construct these functions is crucial for analyzing graphs that depict different behaviors over different domains.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of a function to understand its behavior, such as identifying intervals, slopes, and intercepts. This skill is essential for determining the formula of a piecewise-defined function, as it requires recognizing where the function changes and what mathematical expressions describe each segment of the graph.

Function Formulation

Function formulation is the process of creating a mathematical expression that accurately represents a function's behavior. For piecewise-defined functions, this involves writing separate expressions for each interval and ensuring continuity or identifying points of discontinuity. Mastery of this concept allows one to translate graphical information into precise mathematical language.