Finite Math Examples

$f (x) = 2 x^{7} - x^{5}$

Step 1

Find $f (- x)$ .

Tap for more steps...

Step 1.1

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

$f (- x) = 2 {(- x)}^{7} - {(- x)}^{5}$

Step 1.2

Simplify each term.

Tap for more steps...

Step 1.2.1

Apply the product rule to $- x$ .

$f (- x) = 2 ({(- 1)}^{7} x^{7}) - {(- x)}^{5}$

Step 1.2.2

Raise $- 1$ to the power of $7$ .

$f (- x) = 2 (- x^{7}) - {(- x)}^{5}$

Step 1.2.3

Multiply $- 1$ by $2$ .

$f (- x) = - 2 x^{7} - {(- x)}^{5}$

Step 1.2.4

Apply the product rule to $- x$ .

$f (- x) = - 2 x^{7} - ({(- 1)}^{5} x^{5})$

Step 1.2.5

Multiply $- 1$ by ${(- 1)}^{5}$ by adding the exponents.

Tap for more steps...

Step 1.2.5.1

Move ${(- 1)}^{5}$ .

$f (- x) = - 2 x^{7} + {(- 1)}^{5} \cdot (- 1 x^{5})$

Step 1.2.5.2

Multiply ${(- 1)}^{5}$ by $- 1$ .

Tap for more steps...

Step 1.2.5.2.1

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

$f (- x) = - 2 x^{7} + {(- 1)}^{5} \cdot ((- 1) x^{5})$

Step 1.2.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- x) = - 2 x^{7} + {(- 1)}^{5 + 1} x^{5}$

Step 1.2.5.3

Add $5$ and $1$ .

$f (- x) = - 2 x^{7} + {(- 1)}^{6} x^{5}$

Step 1.2.6

Raise $- 1$ to the power of $6$ .

$f (- x) = - 2 x^{7} + 1 x^{5}$

Step 1.2.7

Multiply $x^{5}$ by $1$ .

$f (- x) = - 2 x^{7} + x^{5}$

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 $- 2 x^{7} + x^{5}$ $\neq$ $2 x^{7} - x^{5}$ , 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 $2 x^{7} - x^{5}$ by $- 1$ .

$- f (x) = - (2 x^{7} - x^{5})$

Step 3.1.2

Apply the distributive property.

$- f (x) = - (2 x^{7}) + x^{5}$

Step 3.1.3

Multiply $2$ by $- 1$ .

$- f (x) = - 2 x^{7} + x^{5}$

Step 3.2

Since $- 2 x^{7} + x^{5} = - 2 x^{7} + x^{5}$ , the function is odd.

The function is odd

Step 4