Algebra Examples

$h (x) = 3 x^{5} + x^{3} - 8 x$

Step 1

Find $f (- x)$ .

Tap for more steps...

Step 1.1

Find $h (- x)$ by substituting $- x$ for all occurrence of $x$ in $h (x)$ .

$h (- x) = 3 {(- x)}^{5} + {(- x)}^{3} - 8 (- x)$

Step 1.2

Simplify each term.

Tap for more steps...

Step 1.2.1

Apply the product rule to $- x$ .

$h (- x) = 3 ({(- 1)}^{5} x^{5}) + {(- x)}^{3} - 8 (- x)$

Step 1.2.2

Raise $- 1$ to the power of $5$ .

$h (- x) = 3 (- x^{5}) + {(- x)}^{3} - 8 (- x)$

Step 1.2.3

Multiply $- 1$ by $3$ .

$h (- x) = - 3 x^{5} + {(- x)}^{3} - 8 (- x)$

Step 1.2.4

Apply the product rule to $- x$ .

$h (- x) = - 3 x^{5} + {(- 1)}^{3} x^{3} - 8 (- x)$

Step 1.2.5

Raise $- 1$ to the power of $3$ .

$h (- x) = - 3 x^{5} - x^{3} - 8 (- x)$

Step 1.2.6

Multiply $- 1$ by $- 8$ .

$h (- x) = - 3 x^{5} - x^{3} + 8 x$

Step 2

A function is even if $f (- x) = f (x)$ .

Tap for more steps...

Step 2.1

Check if $f (- x) = f (x)$ .

Step 2.2

Since $- 3 x^{5} - x^{3} + 8 x$ $\neq$ $3 x^{5} + x^{3} - 8 x$ , the function is not even.

The function is not even

Step 3

A function is odd if $f (- x) = - f (x)$ .

Tap for more steps...

Step 3.1

Find $- f (x)$ .

Tap for more steps...

Step 3.1.1

Multiply $3 x^{5} + x^{3} - 8 x$ by $- 1$ .

$- f (x) = - (3 x^{5} + x^{3} - 8 x)$

Step 3.1.2

Apply the distributive property.

$- f (x) = - (3 x^{5}) - x^{3} - (- 8 x)$

Step 3.1.3

Simplify.

Tap for more steps...

Step 3.1.3.1

Multiply $3$ by $- 1$ .

$- f (x) = - 3 x^{5} - x^{3} - (- 8 x)$

Step 3.1.3.2

Multiply $- 8$ by $- 1$ .

$- f (x) = - 3 x^{5} - x^{3} + 8 x$

Step 3.2

Since $- 3 x^{5} - x^{3} + 8 x = - 3 x^{5} - x^{3} + 8 x$ , the function is odd.

The function is odd

Step 4