Chapter 1.3, Problem 83E

(a)

To determine

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

Answer to Problem 83E

Function is an odd function

Explanation of Solution

Given: Function is $f\left(x\right)=x\sqrt{1-x^2}$

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:

  $f\left(x\right)=x\sqrt{1-x^2}$

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

  $\begin{array}{l}f\left(-x\right)=\left(-x\right)\sqrt{1-{\left(-x\right)}^2}\\ f\left(-x\right)=-x\sqrt{1-x^2}\\ f\left(-x\right)=-\left(x\sqrt{1-x^2}\right)\end{array}$

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

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 83E

Function is an odd function

Explanation of Solution

Given: Function is $f\left(x\right)=x\sqrt{1-x^2}$

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:

  $f\left(x\right)=x\sqrt{1-x^2}$

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

$x$	-1	-0.5	0	0.5	1
$f\left(x\right)=x\sqrt{1-x^2}$	0	-0.433	0	0.433	0

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 83E

Function is an even function

Explanation of Solution

Given: Function is $f\left(x\right)=x\sqrt{1-x^2}$

Calculation:

Function is given as follows:

  $f\left(x\right)=x\sqrt{1-x^2}$

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

x	-1	-0.5	0	0.5	1
$f\left(x\right)$	0	-0.433	0	0.433	0
$f\left(-x\right)$	0	0.433	0	-0.433	0

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

Conclusion:

Hence, function is an odd function