Chapter 1.3, Problem 90E

(a)

To determine

To check function even, odd neither even nor odd algebraically.

Answer to Problem 90E

  $f\left(x\right)=x^5-4x^3$ , is an odd function.

Explanation of Solution

Given: Function is

  $f\left(x\right)=x^5-4x^3$

Calculation:

  $\begin{array}{l}f\left(x\right)=x^5-4x^3\\ f\left(-x\right)={\left(-x\right)}^5-4{\left(-x\right)}^3\\ =-x^5+4x^3\\ =-\left(x^5-4x^3\right)\\ \because f\left(-x\right)=-f\left(x\right)\end{array}$

So, function is an odd function.

(b)

To determine

To check function even, odd neither even nor odd graphically.

Answer to Problem 90E

  $f\left(x\right)=x^5-4x^3$ , is an odd function.

Explanation of Solution

Given: Function is

  $f\left(x\right)=x^5-4x^3$

Observation:

Graph of function is

  

Since graph of function is symmetric about origin, so function is odd function.

(c)

To determine

To check function even, odd neither even nor odd numerically.

Answer to Problem 90E

  $f\left(x\right)=x^5-4x^3$ , is an odd function.

Explanation of Solution

Given: Function is

  $f\left(x\right)=x^5-4x^3$

Calculation:

Tables of values of $x,f\left(x\right)$ and $f\left(-x\right)$ is

  

From table

  $f\left(-x\right)=-f\left(x\right)$

so function is odd function.