Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 1: Functions - Section 1.1 - Functions and Their Graphs - Exercises 1.1 - Page 12: 31

Answer

a. $ y=-x $ when $-1\leq x\lt 0$ $ y=1$ when $0\lt x\leq 1$ $ y=-\frac{1}{2}x+\frac{3}{2}$ when $1\leq x\lt 3$ b. $ y=\frac{1}{2}x $ when $-2\leq x\leq 0$ $ y=-2x+2$ when $0\lt x\leq 1$ $ y=-1$ when $1\lt x\leq 3$

Work Step by Step

a. There are three line segments: (-1,1) to (0,0), (0,1) to (1,1) and (1,1) to (3,0). Using the formula for a line equation passing two known points, we have: Left line segment: $\frac{y-0}{x-0}=\frac{1-0}{-1-0}$ which gives $ y=-x $ when $-1\leq x\lt 0$ Middle line segment: $\frac{y-1}{x-1}=\frac{1-1}{0-1}$ which gives $ y=1$ when $0\lt x\leq 1$ Right line segment: $\frac{y-0}{x-3}=\frac{1-0}{1-3}$ which gives $ y=-\frac{1}{2}x+\frac{3}{2}$ when $1\leq x\lt 3$ b. Similarly, the three line segments can be defined by: (-2,-1) to (0,0), (0,2) to (1,0) and (1,-1) to (3,-1). Left line segment: $\frac{y-0}{x-0}=\frac{-1-0}{-2-0}$ which gives $ y=\frac{1}{2}x $ when $-2\leq x\leq 0$ Middle line segment: $\frac{y-0}{x-1}=\frac{2-0}{0-1}$ which gives $ y=-2x+2$ when $0\lt x\leq 1$ Right line segment: $\frac{y+1}{x-3}=\frac{-1+1}{1-3}$ which gives $ y=-1$ when $1\lt x\leq 3$
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