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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 1 - FunctionsProblema 1.1.30a
Capitolo 1, Problema 1.1.30a

Piecewise-Defined Functions


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


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

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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.
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Piecewise Functions

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.
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Graphing The Derivative

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.
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Properties of Functions
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Piecewise-Defined Functions


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


a. <IMAGE>

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Composition of Functions


Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.


a. y = 2x − 3

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In Exercises 19–32, find the (a) domain and (b) range.

________

𝔂 = 5 - √ x² - 2x - 3

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Functions and Graphs


Graph the following equations and explain why they are not graphs of functions of x.


a. |x| + |y| = 1

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Theory and Examples


The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.


a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)


<IMAGE>

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Functions and Graphs


Graph the following equations and explain why they are not graphs of functions of x.


a. |y| = x

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