Question

select all of the odd functions.
r(x) = x⁷ - 2x³ + 4x
t(x) = 10x⁵ + 7x
m(x) = 2x⁵ - 9x³ + x
j(x) = 4x³ - 7x

Explanation:

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

Step 1: Check \( r(x) = x^7 - 2x^3 + 4x \)

Compute \( r(-x) \):

$$ r(-x) = (-x)^7 - 2(-x)^3 + 4(-x) = -x^7 + 2x^3 - 4x $$

Now, compute \( -r(x) \):

$$ -r(x) = -(x^7 - 2x^3 + 4x) = -x^7 + 2x^3 - 4x $$

Since \( r(-x) = -r(x) \), \( r(x) \) is odd.

Step 2: Check \( t(x) = 10x^5 + 7x \)

Compute \( t(-x) \):

$$ t(-x) = 10(-x)^5 + 7(-x) = -10x^5 - 7x $$

Compute \( -t(x) \):

$$ -t(x) = -(10x^5 + 7x) = -10x^5 - 7x $$

Since \( t(-x) = -t(x) \), \( t(x) \) is odd.

Step 3: Check \( m(x) = 2x^5 - 9x^3 + x \)

Compute \( m(-x) \):

$$ m(-x) = 2(-x)^5 - 9(-x)^3 + (-x) = -2x^5 + 9x^3 - x $$

Compute \( -m(x) \):

$$ -m(x) = -(2x^5 - 9x^3 + x) = -2x^5 + 9x^3 - x $$

Since \( m(-x) = -m(x) \), \( m(x) \) is odd.

Step 4: Check \( j(x) = 4x^3 - 7x \)

Compute \( j(-x) \):

$$ j(-x) = 4(-x)^3 - 7(-x) = -4x^3 + 7x $$

Compute \( -j(x) \):

$$ -j(x) = -(4x^3 - 7x) = -4x^3 + 7x $$

Since \( j(-x) = -j(x) \), \( j(x) \) is odd.

Answer:

All the functions \( r(x) = x^7 - 2x^3 + 4x \), \( t(x) = 10x^5 + 7x \), \( m(x) = 2x^5 - 9x^3 + x \), and \( j(x) = 4x^3 - 7x \) are odd functions. So the selected options (all of them) are correct.