Calculus Examples

$f (x) = 7 x^{2} + 5 x - 4$

Find $f (- x)$ .

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

$f (- x) = 7 {(- x)}^{2} + 5 (- x) - 4$

Simplify each term.

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Apply the product rule to $- x$ .

$f (- x) = 7 ({(- 1)}^{2} x^{2}) + 5 (- x) - 4$

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

$f (- x) = 7 (1 x^{2}) + 5 (- x) - 4$

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

$f (- x) = 7 x^{2} + 5 (- x) - 4$

Multiply $- 1$ by $5$ .

$f (- x) = 7 x^{2} - 5 x - 4$

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

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

$7 x^{2} - 5 x - 4$ $\neq$ $7 x^{2} + 5 x - 4$

Since $7 x^{2} - 5 x - 4$ $\neq$ $7 x^{2} + 5 x - 4$ , 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 $7 x^{2} + 5 x - 4$ by $- 1$ .

$- f (x) = - (7 x^{2} + 5 x - 4)$

Apply the distributive property.

$- f (x) = - (7 x^{2}) - (5 x) + 4$

Simplify.

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Multiply $7$ by $- 1$ .

$- f (x) = - 7 x^{2} - (5 x) + 4$

Multiply $5$ by $- 1$ .

$- f (x) = - 7 x^{2} - 5 x + 4$

Multiply $- 1$ by $- 4$ .

$- f (x) = - 7 x^{2} - 5 x + 4$

Since $7 x^{2} - 5 x - 4$ $\neq$ $- 7 x^{2} - 5 x + 4$ , the function is not odd.

The function is not odd

The function is neither odd nor even

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