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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 1 - Functions문제 1.1.24a
1장, 문제 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

검증된 단계별 안내
1
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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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
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.
추천 영상:
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.
추천 영상:
05:03
Initial Value Problems
관련 실천
교과서 질문

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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교과서 질문

Composition of Functions


Copy and complete the following table.


a. <IMAGE>

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교과서 질문

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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교과서 질문

Functions


In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers.


b. <IMAGE>

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교과서 질문

Piecewise-Defined Functions


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


a. <IMAGE>

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교과서 질문

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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