Precalculus Examples

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

Step 1

Find $f (- x)$ .

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Step 1.1

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

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

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

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

Step 1.2.2

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

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Step 1.2.2.1

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

$f (- x) = 1 + {(- 1)}^{4} \cdot (- 1 x^{4}) + {(- x)}^{2}$

Step 1.2.2.2

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

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Step 1.2.2.2.1

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

$f (- x) = 1 + {(- 1)}^{4} \cdot ((- 1) x^{4}) + {(- x)}^{2}$

Step 1.2.2.2.2

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

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

Step 1.2.2.3

Add $4$ and $1$ .

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

Step 1.2.3

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

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

Step 1.2.4

Apply the product rule to $- x$ .

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

Step 1.2.5

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

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

Step 1.2.6

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

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

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 $1 - x^{4} + x^{2} = 1 - x^{4} + x^{2}$ , the function is even.

The function is even

Step 3