Question

Which of the following is an odd function? f(x) = x3 + 5x2 + x mc023-1.jpg f(x) = x2 + x f(x) = –x

Answer 1

Given the functions
(a) f(x) = x³ + 5x² + x
(b) f(x) = x² + x
(c) f(x) = -x

Function (a)
f(-x) = (-x)³ + 5(-x)² + (-x)
       = -x³ + 5x² - x
       = -(x³ - 5x² + x)
The function is neither even nor odd.

Function (b)
f(-x) = (-x)² + (-x)
        = -(-x² + x)
The function is neither even nor odd.

Function (c)
f(-x) = -(-x)
       = x
       = -f(x)
Because f(-x) = -f(x) the function is odd.

Answer: f(x) = -x is an odd function.

Answer 2

(a). The function $f\left( x \right) = {x^3} + 5{x^2} + x$ is neither even nor odd.

(b).The function $f\left( x \right) = {x^2} + x$ is neither even nor odd.

(c). The function $f\left( x \right) = - x$ is an odd function.

Further explanation:

The function is an even function if $f\left({ - x} \right)=f\left( x \right)$ for all of x.

The function is an odd function if $f\left({ - x} \right)= - f\left( x \right)$ for all of x.

Given:

The functions are as follows,

(a).$f\left( x \right) = {x^3} + 5{x^2} + x$

(b).$f\left( x \right) = {x^2} + x$

(c).$f\left( x \right) = - x$

Explanation:

(a).

Substitute –x for x in function  $f\left( x \right) = {x^3} + 5{x^2} + x.$

$\begin{aligned}f\left({ - x} \right)&={\left({ - x} \right)^3} + 5{\left({ - x}\right)^2}+\left({ - x} \right)\\&=- {x^3} + 5{x^2} - x\\\end{aligned}$

The function is neither even nor odd.

(b).

Substitute –x for x in function  $f\left( x \right) = {x^2} + x.$

$\begin{aligned}f\left( { - x}\right)&= {\left({ - x} \right)^2} + \left({ - x} \right)\\&= {x^2} - x\\\end{aligned}$

(c).

Substitute –x for x in function  $f\left( x \right) = - x.$

$\begin{aligned}f\left({ - x} \right)&= - \left( { - x}\right)\\&=x\\\end{aligned}$

The function is an odd function.

(a). The function $f\left( x \right) = {x^3} + 5{x^2} + x$ is neither even nor odd.

(b).The function $f\left( x \right) = {x^2} + x$ is neither even nor odd.

(c). The function $f\left( x \right)=- x$ is an odd function.

Learn more:

1. Learn more about inverse of the function brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Functions

Keywords: functions, odd functions, even function, polynomial, quadratic function, cubic function, linear function, relations.