Chapter 1.3, Problem 65E

Solution

(a)

To find: The conjecture about symmetry of two basic odd function.

The symmetry of product of two odd function $f(x)=xandg(x)=x^3$ is about y-axis

Given information:

Let one basic function is $f(x)=xandg(x)=x^3$

Multiply these functions we get.

  $f(x)\cdot g(x)=x^4$

  

The symmetry of this function is about y axis as x has even power.

(b)

To find: The conjecture about symmetry of two basic odd function.

The symmetry of product of two odd function $f(x)=x^{2}andg(x)=x^3$ is about origin.

Given information:

Let one basic function is $f(x)=x^{2}andg(x)=\left|x\right|$

Multiply these functions we get.

  $f(x)\cdot g(x)=x^2\cdot \left|x\right|$

  

The symmetry of this function is about origin.

(c)

To find: The conjecture about symmetry of product of an odd function and even function

The symmetry of product of an odd function and an even function $f(x)=x^{2}andg(x)=x^3$ is about origin.

Given information:

Let one basic function is $f(x)=x^{3}andg(x)=x^3$

Multiply these functions we get.

  $f(x)\cdot g(x)=x^2\cdot x^3=x^5$

  

The symmetry of this function is about origin.