Calculus Examples

$f (x) = x^{3} - 2 x$

Find $f (- x)$ .

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

$f (- x) = {(- x)}^{3} - 2 (- x)$

Simplify each term.

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Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{3} x^{3} - 2 (- x)$

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

$f (- x) = - x^{3} - 2 (- x)$

Multiply $- 1$ by $- 2$ .

$f (- x) = - x^{3} + 2 x$

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

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

$- x^{3} + 2 x$ $\neq$ $x^{3} - 2 x$

Since $- x^{3} + 2 x$ $\neq$ $x^{3} - 2 x$ , the function is not even.

The function is not even

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

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Find $- f (x)$ .

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Multiply $x^{3} - 2 x$ by $- 1$ .

$- f (x) = - (x^{3} - 2 x)$

Apply the distributive property.

$- f (x) = - x^{3} - (- 2 x)$

Multiply $- 2$ by $- 1$ .

$- f (x) = - x^{3} + 2 x$

Since $- x^{3} + 2 x = - x^{3} + 2 x$ , the function is odd.

The function is odd

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