Ch. 1 - Functions

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

Hass 15th Edition

Ch. 1 - Functions

Problema 1.1.24a

Capitolo 1, Problema 1.1.24a

Functions and Graphs

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

a. |x| + |y| = 1

Guida verificata passo dopo passo

Step 1: Begin by understanding the equation |x| + |y| = 1. This equation represents a geometric shape known as a diamond or rhombus centered at the origin in the coordinate plane.

Step 2: To graph the equation, consider the absolute value properties. The equation |x| + |y| = 1 can be broken down into four linear equations: x + y = 1, x - y = 1, -x + y = 1, and -x - y = 1. These lines form the boundaries of the diamond shape.

Step 3: Plot these lines on the coordinate plane. Each line will intersect the axes at points that satisfy the equation. For example, x + y = 1 intersects the x-axis at (1, 0) and the y-axis at (0, 1). Repeat this for the other three lines.

Step 4: Connect the points of intersection to form the diamond shape. The resulting graph is symmetric with respect to both the x-axis and y-axis.

Step 5: Explain why this graph is not a function of x. A function of x must pass the vertical line test, meaning that any vertical line drawn through the graph should intersect it at most once. In this case, vertical lines can intersect the diamond at two points, indicating that it is not a function of x.

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Definition of a Function

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for a relation to be a function, it must pass the vertical line test, meaning that no vertical line intersects the graph of the relation at more than one point.

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05:43

Definition of the Definite Integral

Vertical Line Test

The vertical line test is a method used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function of x. This test is crucial for identifying whether a relation can be classified as a function.

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05:13

Slopes of Tangent Lines

Absolute Value Equations

Absolute value equations, such as |x| + |y| = 1, describe geometric shapes in the coordinate plane. This particular equation represents a diamond shape centered at the origin, which can be shown to fail the vertical line test, indicating that for some x-values, there are multiple corresponding y-values, thus confirming it is not a function of x.

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05:03

Initial Value Problems

Pratica correlata

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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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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.

<IMAGE>

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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Composition of Functions

Copy and complete the following table.

a. <IMAGE>

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Find the largest interval on which the given function is increasing.

b. ƒ(x) = (x + 1)⁴

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