HaCkEd By KaSpEr511

Odd and Even Functions

August 10th, 2009

In this post we will discuss problems from 6th Edition Stewart’s Calculus book about odd and even functions on chapter 1, problems 65-70.

Problems

Determine whether $f$ is even, odd, or neither.
65. $f(x)=\frac{x}{x^2+1}$
66. $f(x)=\frac{x^2}{x^4+1}$
67. $f(x)=\frac{x}{x+1}$
68. $f(x)=x|x|$
69. $f(x)=1+3x^2-x^4$
70. $f(x)=1+3x^3-x^5$

Before we do the problems, let us review about odd and even function.

Review

Visually an even function is a function whose graph symmetrical with respect to the $y$ -axis. One example of even function is the function $f(x)=x^2$ .
quadratic
One feature of even function is that if we take two opposite numbers $a$ and $-a$ and compare the value of the function at these points, i.e., comparing $f(a)$ and $f(-a)$ then we have (as we can see from the picture above) $f(-a)=f(a)$ . This property infact is determined the “evennes” of the function in the sense that in order to check whether a function is even, one just need to justify that $f(a)=f(-a)$ for arbitrary $a$ . In case of $f(x)=x^2$ , for arbitrary $a$ we have $f(a)=a^2$ and $f(-a)=(-a)^2$ . So $f(a)=f(-a)$ and we conclude that $f(x)=x^2$ is even.

On the other hand odd functions are the functions whose graphs are symmetrical to the origin. One of the example of an odd function is $f(x)=x^3$ .

cubic
For odd function if we compare the value of $f(x)$ at $-a$ and $a$ we have the relationship that $f(-a)=-f(a)$ as we can see from the picture above. Similar to the case of odd function this property is enough to determine whether a function is odd or not. For example for $f(x)=x^3$ we have $f(-a)=(-a)^3=-a^3$ and $f(a)=a^3$ . Hence $f(-a)=-f(a)$ and we conclude that $f(x)=x^3$ is an odd function.

Remark

Before we do the problem first let me make some remark. Although we can determine whether a function is odd or even from its graph appearance (symmetrical to the $y$ -axis or to the origin) in many cases we can’t always do that. For example you will have hard time trying to graph $f(x)=x^{1000000}$ on your calculator. But it is easy to verify the oddness or the evenness of this function from the algebraic critera given above.

Solution

Let us do problem 67. Consider the value of $f(x)$ at $-a$ and $a$ . We have $f(-a)=\frac{-a}{-a+1}$ and $f(a)=\frac{a}{a+1}$ . In this case $f(-a)\neq f(a)$ and $f(-a)\neq -f(a)$ . Hence $f(x)$ is neither even nor odd.

Now let us take a look problem 68. At $-a$ and $a$ the values of $f(x)$ are $f(-a)=-a|-a|=-a|a|$ (because $|-x|=|x|$ ) and $f(a)=a|a|$ . So here we have $f(-a)=-f(a)$ which means that $f(x)=x|x|$ is an odd function.

All the other problems can be solved by using similar reasoning and I’ll leave to you as exercises.

Categories: 1.1, Algebra, Calculus, Function, Stewart Calculus

Tags: Even Function, Function, Graph, Odd Function, Symmetry Leave a comment

haniifa August 11th, 2009

@Kang Watchmath
Mohon maaf, saya sedang mengajak @Mas Karl Karnadi untuk membuktikan komentar nya mengenai:

Dan btw sama sekali tidak relevan dan bahkan agak misleading untuk anda (maksudnya @Haniifa) menyebutkan kredibilitas orang (mahasiswa, dosen, sampai peserta olimpiade matematika Internasional).

Untuk @Mas Karl Karnadi mari kita bicara baik-baik apapun disiplin ilmu /status sosial saudara disini:

http://haniifa.wordpress.com/2009/08/11/konsep-bilangan-biner-berawal-dari-al-quran/

Wassalam, Haniifa.

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Odd and Even Functions

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Review

Remark

Solution

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