Chapter 1.3, Problem 81E

(a)

To determine

To calculate: To determine whether the function is even, odd or neither algebraically

Answer to Problem 81E

Function is an odd function

Explanation of Solution

Given: Function is $g\left(x\right)=x^3-5x$

Formula Used:

Even Function: Function is even only when $f\left(-x\right)=f\left(x\right)$

Odd Function: Function is odd only when $f\left(-x\right)=-f\left(x\right)$

Calculation:

Function is given as follows:

  $g\left(x\right)=x^3-5x$

Let us find $g\left(-x\right)$ :

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

Since $g\left(-x\right)=-g\left(x\right)$

Thus, function is an odd function

Conclusion:

Hence, the function is an odd function

(b)

To determine

To calculate: To determine whether the function is even, odd or neither graphically

Answer to Problem 81E

Function is an odd function

Explanation of Solution

Given: Function is $g\left(x\right)=x^3-5x$

Formula Used:

Even Function: If the graph of function has symmetry about y-axis as shown below:

  

Odd Function: If the graph of function has symmetry about origin as shown below:

  

Calculation:

Function is given as follows:

  $g\left(x\right)=x^3-5x$

Find the value of $g\left(x\right)$ at different values of $x$

$x$	-2	-1	0	1	2
$g\left(x\right)=x^3-5x$	2	4	0	-4	-2

Plotting the points:

  

Since the graph of function is symmetrical about origin, thus the function is an odd function

Conclusion:

Hence, function is an odd function

(c)

To determine

To calculate: To determine whether the function is even, odd or neither numerically

Answer to Problem 81E

Function is an even function

Explanation of Solution

Given: Function is $g\left(x\right)=x^3-5x$

Calculation:

Function is given as follows:

  $g\left(x\right)=x^3-5x$

Find the value of $g\left(x\right)$ and $g\left(-x\right)$ at different values of $t$

x	-2	-1	0	1	2
$g\left(x\right)$	2	4	0	-4	-2
$g\left(-x\right)$	-2	-4	0	4	2

Since $g\left(-x\right)=-g\left(x\right)$ , thus the function is an odd function

Conclusion:

Hence, function is an odd function