Chapter 1.5, Problem 72E

To determine

Whether the given function $g(x)=x^3-5x$ is even, odd or neither and then describe its symmetry.

Answer to Problem 72E

Given given function $f(x)=x^6-2x^2+3$ is odd and it is symmetric about $x-axis$ and origin.

Explanation of Solution

Given information :

Given function $g(x)=x^3-5x$

As we have $g(x)=x^3-5x$

Replace $x$ by $-x$ in $f(x)$ ,

  $\begin{array}{l}\Rightarrow g(x)={\left(-x\right)}^3-5\left(-x\right)\\ \Rightarrow f(-x)=-x^3+5x\\ \Rightarrow f(-x)=-(x^3-5x)\\ \therefore f(-x)=-f(x)\end{array}$

Thus, given function $g(x)=x^3-5x$ is odd.

Since the function is odd, so symmetry with respect to $x-axis$ .

To check symmetry about $y-axis$ , replace $x$ by $-x$ then

  $\begin{array}{l}g(-x)={\left(-x\right)}^3-5\left(-x\right)\\ g(-x)=-g(x)\end{array}$

This is not equivalent.

Thus, given function is not symmetry about $y-axis$ ,

Similar way, To check symmetry about origin, replace $x$ by $-x$ and $y$ by $-y$ then

  $\begin{array}{l}\Rightarrow -y={\left(-x\right)}^3-5\left(-x\right)\\ \Rightarrow -y=-x^3+5x\\ \Rightarrow -y=-g(x)\\ \therefore y=g(x)\end{array}$

This is equivalent.

Thus, given function is symmetry about origin,

Hence, given function $f(x)=x^6-2x^2+3$ is odd and it is symmetric about $x-axis$ and origin.