Precalculus Examples

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

Step 1

Find $f (- x)$ .

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Step 1.1

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

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

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $- (- x)$ .

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Step 1.2.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.4.2

Multiply $x$ by $1$ .

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

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 $e^{x^{4} + x}$ $\neq$ $e^{x^{4} - x}$ , the function is not even.

The function is not even

Step 3

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

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Step 3.1

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

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

Step 3.2

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

The function is not odd

Step 4

The function is neither odd nor even

Step 5