Question

select all of the odd functions.
$b(x) = -8x^2 - 3$
$n(x) = 2x^5 + 7x^3 + 8x$
$h(x) = -9x^5 + 4x$
$q(x) = 5x^7 - 3x^3 - 4x$

Explanation:

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

Step 1: Analyze \( b(x) = -8x^2 - 3 \)

Compute \( b(-x) \):
\( b(-x) = -8(-x)^2 - 3 = -8x^2 - 3 \)
Now, compute \( -b(x) \):
\( -b(x) = -(-8x^2 - 3) = 8x^2 + 3 \)
Since \( b(-x)
eq -b(x) \), \( b(x) \) is not an odd function.

Step 2: Analyze \( n(x) = 2x^5 + 7x^3 + 8x \)

Compute \( n(-x) \):
\( n(-x) = 2(-x)^5 + 7(-x)^3 + 8(-x) = -2x^5 - 7x^3 - 8x \)
Now, compute \( -n(x) \):
\( -n(x) = -(2x^5 + 7x^3 + 8x) = -2x^5 - 7x^3 - 8x \)
Since \( n(-x) = -n(x) \), \( n(x) \) is an odd function.

Step 3: Analyze \( h(x) = -9x^5 + 4x \)

Compute \( h(-x) \):
\( h(-x) = -9(-x)^5 + 4(-x) = 9x^5 - 4x \)
Now, compute \( -h(x) \):
\( -h(x) = -(-9x^5 + 4x) = 9x^5 - 4x \)
Since \( h(-x) = -h(x) \), \( h(x) \) is an odd function.

Step 4: Analyze \( q(x) = 5x^7 - 3x^3 - 4x \)

Compute \( q(-x) \):
\( q(-x) = 5(-x)^7 - 3(-x)^3 - 4(-x) = -5x^7 + 3x^3 + 4x \)
Now, compute \( -q(x) \):
\( -q(x) = -(5x^7 - 3x^3 - 4x) = -5x^7 + 3x^3 + 4x \)
Since \( q(-x) = -q(x) \), \( q(x) \) is an odd function.

Answer:

The odd functions are:

• \( n(x) = 2x^5 + 7x^3 + 8x \)
• \( h(x) = -9x^5 + 4x \)
• \( q(x) = 5x^7 - 3x^3 - 4x \)