Algebra Examples

$f (x) = x^{4} - x^{12} + 1$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = {(- x)}^{4} - {(- x)}^{12} + 1$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{4} x^{4} - {(- x)}^{12} + 1$

Step 1.2.2

Raise $- 1$ to the power of $4$ .

$f (- x) = 1 x^{4} - {(- x)}^{12} + 1$

Step 1.2.3

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

$f (- x) = x^{4} - {(- x)}^{12} + 1$

Step 1.2.4

Apply the product rule to $- x$ .

$f (- x) = x^{4} - ({(- 1)}^{12} x^{12}) + 1$

Step 1.2.5

Multiply $- 1$ by ${(- 1)}^{12}$ by adding the exponents.

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Step 1.2.5.1

Move ${(- 1)}^{12}$ .

$f (- x) = x^{4} + {(- 1)}^{12} \cdot (- 1 x^{12}) + 1$

Step 1.2.5.2

Multiply ${(- 1)}^{12}$ by $- 1$ .

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Step 1.2.5.2.1

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

$f (- x) = x^{4} + {(- 1)}^{12} \cdot ((- 1) x^{12}) + 1$

Step 1.2.5.2.2

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

$f (- x) = x^{4} + {(- 1)}^{12 + 1} x^{12} + 1$

Step 1.2.5.3

Add $12$ and $1$ .

$f (- x) = x^{4} + {(- 1)}^{13} x^{12} + 1$

Step 1.2.6

Raise $- 1$ to the power of $13$ .

$f (- x) = x^{4} - x^{12} + 1$

Step 2

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

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Step 2.1

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

Step 2.2

Since $x^{4} - x^{12} + 1 = x^{4} - x^{12} + 1$ , the function is even.

The function is even

Step 3