Chapter 1.3, Problem 84E

(a)

To determine

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

Answer to Problem 84E

Function is neither even nor odd

Explanation of Solution

  

Given: Function is $f\left(x\right)=x\sqrt{x+5}$

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:

  $f\left(x\right)=x\sqrt{x+5}$

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

  $\begin{array}{l}f\left(-x\right)=\left(-x\right)\sqrt{\left(-x\right)+5}\\ f\left(-x\right)=-x\sqrt{5-x}\end{array}$

Since $f\left(-x\right)\ne f\left(x\right)$ and $f\left(-x\right)\ne -f\left(x\right)$

Thus, function is neither even nor odd

Conclusion:

Hence, the function is neither even nor odd

(b)

To determine

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

Answer to Problem 84E

Function is neither even nor odd

Explanation of Solution

  

Given: Function is $f\left(x\right)=x\sqrt{x+5}$

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:

  $f\left(x\right)=x\sqrt{x+5}$

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

x	-5	-4	-1	0	4
$f\left(x\right)=x\sqrt{x+5}$	0	-4	-2	0	12

Plotting the points:

  

Since the graph of function is neither symmetrical about y-axis not about origin, thus the function is neither even nor odd

Conclusion:

Hence, function is neither even nor odd

(c)

To determine

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

Answer to Problem 84E

Function is neither even nor odd

Explanation of Solution

  

Given: Function is $f\left(x\right)=x\sqrt{x+5}$

Calculation:

Function is given as follows:

  $f\left(x\right)=x\sqrt{x+5}$

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

$x$	-5	-4	-1	0	4
$f\left(x\right)$	0	-4	-2	0	12
$f\left(-x\right)$	15.8	12	2.44	0	-4

Since $f\left(-x\right)\ne f\left(x\right)$ and $f\left(-x\right)\ne -f\left(x\right)$ , thus the function is neither even nor odd

Conclusion:

Hence, function is neither even nor odd