Precalculus Examples

$y = x^{5} - 4 x$

Step 1

Write $y = x^{5} - 4 x$ as a function.

$f (x) = x^{5} - 4 x$

Step 2

Find $f (- x)$ .

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Step 2.1

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

$f (- x) = {(- x)}^{5} - 4 (- x)$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{5} x^{5} - 4 (- x)$

Step 2.2.2

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

$f (- x) = - x^{5} - 4 (- x)$

Step 2.2.3

Multiply $- 1$ by $- 4$ .

$f (- x) = - x^{5} + 4 x$

Step 3

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

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Step 3.1

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

Step 3.2

Since $- x^{5} + 4 x$ $\neq$ $x^{5} - 4 x$ , the function is not even.

The function is not even

Step 4

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

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Step 4.1

Find $- f (x)$ .

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Step 4.1.1

Multiply $x^{5} - 4 x$ by $- 1$ .

$- f (x) = - (x^{5} - 4 x)$

Step 4.1.2

Apply the distributive property.

$- f (x) = - x^{5} - (- 4 x)$

Step 4.1.3

Multiply $- 4$ by $- 1$ .

$- f (x) = - x^{5} + 4 x$

Step 4.2

Since $- x^{5} + 4 x = - x^{5} + 4 x$ , the function is odd.

The function is odd

Step 5