Question

select all of the odd functions.
$g(x) = -x^3 + 7$
$w(x) = x^4 + 2x^2 - 9$
$p(x) = -x^5 + 2x^3 + 8x$
$m(x) = -4x^5 + x^3$

Explanation:

To determine if a function \( f(x) \) is odd, we use the definition: \( f(-x) = -f(x) \) for all \( x \) in the domain. Let's analyze each function:

For \( g(x) = -x^3 + 7 \)

• Compute \( g(-x) \):

\( g(-x) = -(-x)^3 + 7 = -(-x^3) + 7 = x^3 + 7 \)

• Compute \( -g(x) \):

\( -g(x) = -(-x^3 + 7) = x^3 - 7 \)
Since \( g(-x)
eq -g(x) \), \( g(x) \) is not odd.

For \( w(x) = x^4 + 2x^2 - 9 \)

• Compute \( w(-x) \):

\( w(-x) = (-x)^4 + 2(-x)^2 - 9 = x^4 + 2x^2 - 9 \)

• Compute \( -w(x) \):

\( -w(x) = -(x^4 + 2x^2 - 9) = -x^4 - 2x^2 + 9 \)
Since \( w(-x)
eq -w(x) \), \( w(x) \) is not odd (it is even, as \( w(-x) = w(x) \)).

For \( p(x) = -x^5 + 2x^3 + 8x \)

• Compute \( p(-x) \):

\( p(-x) = -(-x)^5 + 2(-x)^3 + 8(-x) = -(-x^5) + 2(-x^3) - 8x = x^5 - 2x^3 - 8x \)

• Compute \( -p(x) \):

\( -p(x) = -(-x^5 + 2x^3 + 8x) = x^5 - 2x^3 - 8x \)
Since \( p(-x) = -p(x) \), \( p(x) \) is odd.

For \( m(x) = -4x^5 + x^3 \)

• Compute \( m(-x) \):

\( m(-x) = -4(-x)^5 + (-x)^3 = -4(-x^5) + (-x^3) = 4x^5 - x^3 \)

• Compute \( -m(x) \):

\( -m(x) = -(-4x^5 + x^3) = 4x^5 - x^3 \)
Since \( m(-x) = -m(x) \), \( m(x) \) is odd.

Correct Selections:

The odd functions are \( \boldsymbol{p(x) = -x^5 + 2x^3 + 8x} \) and \( \boldsymbol{m(x) = -4x^5 + x^3} \). The initial selections for \( g(x) \) and \( w(x) \) were incorrect.

Answer:

• \( p(x) = -x^5 + 2x^3 + 8x \) (select this)
• \( m(x) = -4x^5 + x^3 \) (select this)
• Deselect \( g(x) = -x^3 + 7 \) and \( w(x) = x^4 + 2x^2 - 9 \).