Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 1 - Functions

Problema 1.1.31a

Capitolo 1, Problema 1.1.31a

Piecewise-Defined Functions

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

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Identify the different segments of the graph. A piecewise function is defined by different expressions over different intervals. Look at the graph and note where the function changes its behavior.

Determine the type of function for each segment. Common types include linear, quadratic, or constant functions. Analyze the graph to see if the segments are straight lines, curves, or flat lines.

Find the equations for each segment. For linear segments, use the slope-intercept form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. For quadratic segments, use the standard form $ y = ax^2 + bx + c $.

Specify the domain for each piece of the function. The domain is the interval over which each segment is valid. Use the x-values where the function changes to define these intervals.

Combine the equations and domains into a piecewise function. Write the function using the format: $ f(x) = \begin{cases} \text{expression 1,} & \text{if } \text{domain 1} \\ \text{expression 2,} & \text{if } \text{domain 2} \\ \end{cases} $.

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Piecewise-Defined Functions

A piecewise-defined function is a function composed of multiple sub-functions, each of which applies to a specific interval of the domain. These functions are defined by different expressions based on the input value. Understanding how to interpret and construct these functions is crucial for analyzing graphs that change behavior at certain points.

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05:36

Piecewise Functions

Graph Interpretation

Graph interpretation involves analyzing a 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 requires recognizing where the function changes and what expressions describe each segment.

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06:15

Graphing The Derivative

Function Continuity and Discontinuity

Continuity in a function means that the graph is unbroken, while discontinuity indicates a jump, hole, or asymptote. Identifying these features in a graph is vital for writing piecewise functions, as they often dictate where one piece ends and another begins, and whether the function is defined at those transition points.

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05:34

Intro to Continuity

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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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Industrial costs A power plant sits next to a river where the river is 800 ft wide. Laying a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs \$180 per foot across the river and \$100 per foot along the land.

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b. Generate a table of values to determine whether the least expensive location for point Q is less than 2000 ft or greater than 2000 ft from point P.

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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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Domanda del libro di testo

Composition of Functions

Copy and complete the following table.

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