Algebra Examples

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

Find $f (- x)$ .

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

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

Simplify each term.

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Apply the product rule to $- x$ .

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

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

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

Apply the product rule to $- x$ .

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

Multiply $- 1$ by ${(- 1)}^{4}$ by adding the exponents.

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Move ${(- 1)}^{4}$ .

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

Multiply ${(- 1)}^{4}$ by $- 1$ .

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

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

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

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

Add $4$ and $1$ .

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

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

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

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

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

$- x^{5} - x^{4}$ $\neq$ $x^{5} - x^{4}$

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

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

Apply the distributive property.

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

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

The function is not odd

The function is neither odd nor even