Chapter 1.3, Problem 80E

(a)

To determine

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

Answer to Problem 80E

Function is an even function

Explanation of Solution

  

Given: Function is $f\left(x\right)=x^6-2x^2+3$

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:

  $f\left(x\right)=x^6-2x^2+3$

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

  $\begin{array}{l}f\left(-x\right)={\left(-x\right)}^6-2{\left(-x\right)}^2+3\\ f\left(-x\right)=x^6-2x^2+3\end{array}$

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

Thus, function is an even function

Conclusion:

Hence, the function is an even function

(b)

To determine

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

Answer to Problem 80E

Function is an even function

Explanation of Solution

  

Given: Function is $f\left(x\right)=x^6-2x^2+3$

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:

  $f\left(x\right)=x^6-2x^2+3$

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

$x$	-2	-1	0	1	2
$f\left(x\right)=x^6-2x^2+3$	59	2	3	2	59

Plotting the points:

  

Since the graph of function is symmetrical about y-axis, thus the function is an even function

Conclusion:

Hence, function is an even function

(c)

To determine

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

Answer to Problem 80E

Function is an even function

Explanation of Solution

  

Given: Function is $f\left(x\right)=x^6-2x^2+3$

Calculation:

Function is given as follows:

  $f\left(x\right)=x^6-2x^2+3$

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

t	-2	-1	0	1	2
$f\left(x\right)$	59	2	3	2	59
$f\left(-x\right)$	59	2	3	2	59

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

Conclusion:

Hence, function is an even function