College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 3 - Section 3.4 - Library of Functions; Piecewise-defined Function - 3.4 Assess Your Understanding - Page 247: 70

Answer

1. It is a function. 2. Domain: $\mathbb{R}$ 3. Range: $\{0,1\}$ 4. x-intercepts : all irrational numbers 5. y-intercept: 1 6. Even. 7. Cannot be graphed accurately. See explanation below.

Work Step by Step

1. It is a function. For every input (real number x), a single value is defined as f(x). 2. Domain: $\mathbb{R}$ (all real numbers have a function value) 3. Range: $\{0,1\}$ (only one of two values are awarded to each real number) 4. x-intercepts : all irrational numbers (for them, f(x)=0) 5. y-intercepts: 1 (f(0)=1) 6. If x is rational, so is -x $\Rightarrow$ f(-x)=f(x) If x is irrational, so is -x $\Rightarrow$ f(-x)=f(x) So, the function is even. 7. The graph would resemble two parallel lines, $y=1$ and $y=0$ BUT the lines are not accurate graphs, as each has an infinite number of points excluded, and we can't plot the open dots, because in between any two rational numbers there is an infinite number of both rational and irrational numbers. The line $y=1 $has all points (x,1) excluded where x is an irrational number, and the line $y=0 $ has all points (x,0) excluded where x is a rational number. In other words, we can't accurately graph this function. The best we can do is cluster points along these lines to give an indication of the general layout.
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