Question

Find an if the function f(x) = x - x³.​

Answer 1

Answer:

The function f(x) is odd, if:

f(−x)=−f(x)

and even, if:

f(−x)=f(x)

Given that

f(x)=∣x∣−x

3

f(−x)=∣−x∣−(−x)

3

=∣−x∣+x

3

Since ∣x∣ is an even function,

∣−x∣=∣x∣

∴f(−x)=∣x∣+x

3

Also,

−f(x)=−∣x∣+x

3

Hence,

f(−x)



=f(x)

and

f(−x)



=−f(x)

f(x) is neither odd nor even

Answer 2

Given :

An equation $f(x)=x-x^{3}$.

To find :

The function of the given equation.

Step-by-step explanation:

• The function can be found by following these steps.
• If the power is odd, while substituting a negative number, it leads to a negative number.
• If the power is odd and the substituted number is a positive number, it leads to a positive number.
• Incase of the power being even, irrespective of the number substituted being negative or positive, the resultant value is going to be positive.
• In this case, the function becomes greater than $0$ when $x$ becomes negative.
• The function becomes lesser than $0$ when $x$ becomes positive.
• Therefore, the function will be

         $f(x)=x-x^{3} >0$ when $x<0$

• The other function would be

        $f(x)=x-x^{3} <0$ when $x>0$

• In case of $0$

        $f(x)=0$ when $x=0$

Final answer :

The functions would be $f(x)=x-x^{3} >0$ when $x<0$ ,  $f(x)=x-x^{3} <0$ when $x>0$ , $f(x)=0$ when $x=0$.