Calculus Examples

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

Find $f (- x)$ .

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

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

Simplify each term.

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Apply the product rule to $- x$ .

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

Raise $- 1$ to the power of $2$ .

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

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

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

Multiply $- 1$ by $- 5$ .

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

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

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

$x^{2} + 5 x + 6$ $\neq$ $x^{2} - 5 x + 6$

Since $x^{2} + 5 x + 6$ $\neq$ $x^{2} - 5 x + 6$ , 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 $x^{2} - 5 x + 6$ by $- 1$ .

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

Apply the distributive property.

$- f (x) = - x^{2} - (- 5 x) - 1 \cdot 6$

Simplify.

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Multiply $- 5$ by $- 1$ .

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

Multiply $- 1$ by $6$ .

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

Since $x^{2} + 5 x + 6$ $\neq$ $- x^{2} + 5 x - 6$ , the function is not odd.

The function is not odd

The function is neither odd nor even

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