Question

select all of the odd functions.
k(x) = -3x⁵ - 4x³ + x
v(x) = -x⁴ + 5x² + 2
n(x) = 3x⁵ - 2x
d(x) = -7x² + 4x

Explanation:

Step1: Recall the definition of an odd function

A function \( f(x) \) is odd if \( f(-x)=-f(x) \) for all \( x \) in the domain.

Step2: Test \( k(x) = -3x^5 - 4x^3 + x \)

Calculate \( k(-x) \):
\( k(-x)=-3(-x)^5 - 4(-x)^3+(-x) \)
\( = -3(-x^5)-4(-x^3)-x \)
\( = 3x^5 + 4x^3 - x \)
Now, calculate \( -k(x) \):
\( -k(x)=-(-3x^5 - 4x^3 + x)=3x^5 + 4x^3 - x \)
Since \( k(-x)=-k(x) \), \( k(x) \) is odd.

Step3: Test \( v(x) = -x^4 + 5x^2 + 2 \)

Calculate \( v(-x) \):
\( v(-x)=-(-x)^4 + 5(-x)^2 + 2 \)
\( = -x^4 + 5x^2 + 2 \)
Calculate \( -v(x) \):
\( -v(x)=-(-x^4 + 5x^2 + 2)=x^4 - 5x^2 - 2 \)
Since \( v(-x)
eq -v(x) \), \( v(x) \) is even (not odd).

Step4: Test \( n(x) = 3x^5 - 2x \)

Calculate \( n(-x) \):
\( n(-x)=3(-x)^5 - 2(-x) \)
\( = 3(-x^5)+2x=-3x^5 + 2x \)
Calculate \( -n(x) \):
\( -n(x)=-(3x^5 - 2x)=-3x^5 + 2x \)
Since \( n(-x)=-n(x) \), \( n(x) \) is odd.

Step5: Test \( d(x) = -7x^2 + 4x \)

Calculate \( d(-x) \):
\( d(-x)=-7(-x)^2 + 4(-x) \)
\( = -7x^2 - 4x \)
Calculate \( -d(x) \):
\( -d(x)=-(-7x^2 + 4x)=7x^2 - 4x \)
Since \( d(-x)
eq -d(x) \), \( d(x) \) is not odd.

Answer:

The odd functions are \( k(x) = -3x^5 - 4x^3 + x \) and \( n(x) = 3x^5 - 2x \) (i.e., the first and third options).