Question

select all of the odd functions.
k(x) = 2x⁴ - x² + 5
g(x) = x³ - 4x
h(x) = 3x² + 1
z(x) = x² + x

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: Check \( k(x) = 2x^{4}-x^{2}+5 \)

Calculate \( k(-x) \):
\( k(-x)=2(-x)^{4}-(-x)^{2}+5 = 2x^{4}-x^{2}+5 \)
Now, \( -k(x)=-(2x^{4}-x^{2}+5)=-2x^{4}+x^{2}-5 \)
Since \( k(-x)
eq -k(x) \), \( k(x) \) is not odd.

Step3: Check \( g(x)=x^{3}-4x \)

Calculate \( g(-x) \):
\( g(-x)=(-x)^{3}-4(-x)=-x^{3}+4x \)
Now, \( -g(x)=-(x^{3}-4x)=-x^{3}+4x \)
Since \( g(-x)=-g(x) \), \( g(x) \) is odd.

Step4: Check \( h(x)=3x^{2}+1 \)

Calculate \( h(-x) \):
\( h(-x)=3(-x)^{2}+1 = 3x^{2}+1 \)
Now, \( -h(x)=-(3x^{2}+1)=-3x^{2}-1 \)
Since \( h(-x)
eq -h(x) \), \( h(x) \) is not odd.

Step5: Check \( z(x)=x^{2}+x \)

Calculate \( z(-x) \):
\( z(-x)=(-x)^{2}+(-x)=x^{2}-x \)
Now, \( -z(x)=-(x^{2}+x)=-x^{2}-x \)
Since \( z(-x)
eq -z(x) \), \( z(x) \) is not odd.

Answer:

\( g(x) = x^{3}-4x \)