CK-12 Calculus Study Guide - Even and Odd Functions Sanjeev Narayanaswamy Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) AUTHOR Sanjeev Narayanaswamy To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2015 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Com- mons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/about/ terms-of-use. Printed: April 6, 2015 www.ck12.org Chapter 1. CK-12 Calculus Study Guide - Even and Odd Functions CHAPTER 1 CK-12 Calculus Study Guide - Even and Odd Functions Important Points - Even and Odd Functions Let f (x) represent any function. f (x) can be classified as either even, odd, or neither. Even Function • f (x) is defined to be an even function if and only if it satisfies f (−x) = f (x) • f (x) is symmetric about the y axis. i.e. f (x) = x2 Odd Function • f (x) is defined to be an odd function if it satisfies f (−x) = − f (x) • f (x) is rotationally symmetric about the origin. i.e. f (x) = x3 Neither Odd nor Even • A function is neither odd nor even if it doesn’t satisfy the conditions for odd or even functions. • A quick test for this is if f (−x) 6= f (x) and f (−x) 6= − f (x). Useful Properties • sin(−x) = −sin(x) • cos(−x) = cos(x) • j−xj= jxj • If f (x) = xn, where n is a natural number (positive, non-zero integer), then f (x) is even if n is even and odd if n is odd. Example 1 Question: Which of the following is NOT a property of an even function? Pick the best answer. a) symmetric about y axis b) f (x) = − f (x) c) symmetric about x axis d) a and b e) b and c f) none of the above Analysis: An even function must satisfy f (−x) = f (x).This also implies symmetry about the y-axis. From the options given, the properties listed in b) and c) are both not properties of even functions. Answer: e) 1 www.ck12.org Example 2 Question: Classify the following function as even, odd, or neither: f (x) = (2x2 + 5x)2 Analysis: The first step in the algebraic approach to classifying functions as even, odd, or neither is to replace x with −x. Then we need to see if f (−x) = f (x) or f (−x) = − f (x) is satisfied. f (x) = (2x2 + 5x)2 f (−x) = (2(−x)2 + 5(−x))2 = (2(−x)2 + 5(−x))2 = (2(x)2 − 5(x))2 = (2x2 − 5x)2 − f (x) = −(2x2 + 5x)2 In this analysis, we see that f (−x) 6= − f (x) and f (−x) 6= f (x). Answer: Neither Below is the graph of the function for reference: Example 3 Question: Classify the following function as even, odd, or neither: jxj f (x) = (j2xj+1) 2 www.ck12.org Chapter 1. CK-12 Calculus Study Guide - Even and Odd Functions Analysis: The first step in the algebraic approach to classifying functions as even, odd, or neither is to replace x with −x. Then we need to see if f (−x) = f (x) or f (−x) = − f (x) is satisfied. jxj f (x) = (j2xj+1) j−xj f (−x) = (j2(−x)j+1) j−xj = (j−2xj+1) jxj = (j2xj+1) In this analysis, we have used the property that jxj= j−xj. We see that f (x) = f (−x). Answer: Even Below is the graph of the function for reference: Example 4 Question: Classify the following function as even, odd, or neither: f (x) = jsin(x)j Analysis: Check if f (−x) = f (x) or f (−x) = − f (x) is satisfied. f (x) = jsin(x)j f (−x) = jsin(−x)j = j−sin(x)j = jsin(x)j In this analysis, we have used the property that sin(−x) = −sin(x) and jxj= j−xj. We see that f (x) = f (−x). Answer: Even Below is the graph of the function for reference: 3 www.ck12.org Example 5 Question: Classify the following function as even, odd, or neither: jsin(x)j f (x) = sin(x) Analysis: Check if f (−x) = f (x) or f (−x) = − f (x) is satisfied. jsin(x)j f (x) = sin(x) jsin(−x)j f (−x) = sin(−x) jsin(−x)j = sin(−x) j−sin(x)j = −sin(x) −jsin(x)j = sin(x) In this analysis, we have used the property that sin(−x) = −sin(x) and jxj= j−xj. We see that f (−x) = − f (x). Answer: Odd Below is the graph of the function for reference: 4 www.ck12.org Chapter 1. CK-12 Calculus Study Guide - Even and Odd Functions Note that the given function is not defined for x = np, where n is any integer. Example 6 Question: Classify the following function as even, odd, or neither: f (x) = ex Analysis: Check if f (−x) = f (x) or f (−x) = − f (x) is satisfied. f (x) = ex f (−x) = e−x In this analysis, we see that f (−x) 6= − f (x) and f (−x) 6= f (x). Answer: Neither Below is the graph of the function for reference: 5 www.ck12.org Example 7 Question: Classify the following function as even, odd, or neither: q f (x) = jx3j Analysis: Check if f (−x) = f (x) or f (−x) = − f (x) is satisfied. q f (x) = jx3j q f (−x) = j(−x)3j q = j(−1)3x3j q = j−x3j q = jx3j In this analysis, we have used the property that jxj= j−xj. We see that f (x) = f (−x). Answer: Even Below is the graph of the function for reference: Example 8 Question: Classify the following function as even, odd, or neither: f (x) = cos(x) + sin(x) 6 www.ck12.org Chapter 1. CK-12 Calculus Study Guide - Even and Odd Functions Analysis: Check if f (−x) = f (x) or f (−x) = − f (x) is satisfied. f (x) = cos(x) + sin(x) f (−x) = cos(−x) + sin(−x) = cos(−x) + sin(−x) = cos(x) + (−sin(x)) In this analysis, we have used the property that cos(−x) = cos(x) and sin(−x) = −sin(x). We see that f (−x) 6= − f (x) and f (−x) 6= f (x). Answer: Neither Below is the graph of the function for reference: Example 9 Question: Classify the following function as even, odd, or neither: f (x) = sin[sin(x)] Analysis: Check if f (−x) = f (x) or f (−x) = − f (x) is satisfied. f (x) = sin[sin(x)] f (−x) = sin[sin(−x)] = sin[−sin(x)] = −sin[sin(x)] 7 www.ck12.org In this analysis, we have used the property that sin(−x) = −sin(x) twice to bring out the negative sign. We see that f (−x) = − f (x). Answer: Odd Below is the graph of the function for reference: Example 10 Question: Classify the following function as even, odd, or neither: jxj sin(ee ) f (x) = pcos(x) Analysis: Check if f (−x) = f (x) or f (−x) = − f (x) is satisfied. jxj sin(ee ) f (x) = pcos(x) j−xj sin(ee ) f (−x) = pcos(−x) jxj sin(ee ) = pcos(x) To analyze this very complicated function, we have simply used the property that jxj= j−xj and cos(−x) = cos(x). We see that f (−x) = − f (x). Answer: Even Below is the graph of the function for reference: 8 www.ck12.org Chapter 1. CK-12 Calculus Study Guide - Even and Odd Functions 9.