Chapter 4, Problem 7CA

(a)

To determine

To find: Whether the given function $f\left(x\right)=3x^5$ is even or odd.

Answer to Problem 7CA

The required solution is odd.

Explanation of Solution

Given information:

The given equation is:

  $f\left(x\right)=3x^5$

Consider the given expression:

  $f\left(x\right)=3x^5$

Replace $x$ by $-x$

  $\begin{array}{c}f\left(-x\right)=3{\left(-x\right)}^5\\ f\left(-x\right)=-3x^5\\ f\left(-x\right)=-f\left(x\right)\end{array}$

It is seen that $f\left(-x\right)=-f\left(x\right)$ .

Hence, the required solution is odd.

(b)

To determine

To find: Whether the given function $f\left(x\right)=4x^3+8x$ is even or odd.

Answer to Problem 7CA

The required solution is odd.

Explanation of Solution

Given information:

The given equation is:

  $f\left(x\right)=4x^3+8x$

Consider the given expression:

  $f\left(x\right)=4x^3+8x$

Replace $x$ by $-x$

  $\begin{array}{c}f\left(-x\right)=4{\left(-x\right)}^3+8\left(-x\right)\\ =-4x^3-8x\\ =-\left(4x^3+8x\right)\\ =-f\left(x\right)\end{array}$

It is seen that $f\left(-x\right)=-f\left(x\right)$ .

Hence, the required solution is odd.

(c)

To determine

To find: Whether the given function $f\left(x\right)=3x^3+12x^2+1$ is even or odd.

Answer to Problem 7CA

The required solution is neither odd nor even.

Explanation of Solution

Given information:

The given equation is:

  $f\left(x\right)=3x^3+12x^2+1$

Consider the given expression:

  $f\left(x\right)=3x^3+12x^2+1$

Replace $x$ by $-x$

  $\begin{array}{l}f\left(-x\right)=3{\left(-x\right)}^3+12{\left(-x\right)}^2+1\\ f\left(-x\right)=-\left(3x^3-12x^2-1\right)\end{array}$

It is seen that $f\left(-x\right)\cancel{=}-f\left(x\right)$ .

Hence, the required solution is neither odd nor even.

(d)

To determine

To find: Whether the given function $f\left(x\right)=2x^4$ is even or odd.

Answer to Problem 7CA

The required solution is even.

Explanation of Solution

Given information:

The given equation is:

  $f\left(x\right)=2x^4$

Consider the given expression:

  $f\left(x\right)=2x^4$

Replace $x$ by $-x$

  $\begin{array}{c}f\left(-x\right)=2{\left(-x\right)}^4\\ f\left(-x\right)=2x^4\\ f\left(-x\right)=f\left(x\right)\end{array}$

It is seen that $f\left(-x\right)=f\left(x\right)$ .

Hence, the required solution is even.

(e)

To determine

To find: Whether the given function $f\left(x\right)=x^{11}-x^7$ is even or odd.

Answer to Problem 7CA

The required solution is odd.

Explanation of Solution

Given information:

The given equation is:

  $f\left(x\right)=x^{11}-x^7$

Consider the given expression:

  $f\left(x\right)=x^{11}-x^7$

Replace $x$ by $-x$

  $\begin{array}{c}f\left(-x\right)={\left(-x\right)}^{11}-{\left(-x\right)}^7\\ f\left(-x\right)=-\left(x^{11}-x^7\right)\\ f\left(-x\right)=-f\left(x\right)\end{array}$

It is seen that $f\left(-x\right)=-f\left(x\right)$ .

Hence, the required solution is odd.

(f)

To determine

To find: Whether the given function $f\left(x\right)=2x^8+4x^4+x^2-5$ is even or odd.

Answer to Problem 7CA

The required solution is even.

Explanation of Solution

Given information:

The given equation is:

  $f\left(x\right)=2x^8+4x^4+x^2-5$

Consider the given expression:

  $f\left(x\right)=2x^8+4x^4+x^2-5$

Replace $x$ by $-x$

  $\begin{array}{l}f\left(x\right)=2{\left(-x\right)}^8+4{\left(-x\right)}^4+{\left(-x\right)}^2-5\\ f\left(x\right)=2x^8+4x^4+x^2-5\\ f\left(x\right)=f\left(x\right)\end{array}$

It is seen that $f\left(-x\right)=f\left(x\right)$ .

Hence, the required solution is even.