Calculus Examples

$f (x) = 3 x - 4 + 2 x^{2}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = 3 (- x) - 4 + 2 {(- x)}^{2}$

Step 1.2

Simplify each term.

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Step 1.2.1

Multiply $- 1$ by $3$ .

$f (- x) = - 3 x - 4 + 2 {(- x)}^{2}$

Step 1.2.2

Apply the product rule to $- x$ .

$f (- x) = - 3 x - 4 + 2 ({(- 1)}^{2} x^{2})$

Step 1.2.3

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

$f (- x) = - 3 x - 4 + 2 (1 x^{2})$

Step 1.2.4

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

$f (- x) = - 3 x - 4 + 2 x^{2}$

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 $- 3 x - 4 + 2 x^{2}$ $\neq$ $3 x - 4 + 2 x^{2}$ , 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

Find $- f (x)$ .

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Step 3.1.1

Multiply $3 x - 4 + 2 x^{2}$ by $- 1$ .

$- f (x) = - (3 x - 4 + 2 x^{2})$

Step 3.1.2

Apply the distributive property.

$- f (x) = - (3 x) + 4 - (2 x^{2})$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply $3$ by $- 1$ .

$- f (x) = - 3 x + 4 - (2 x^{2})$

Step 3.1.3.2

Multiply $- 1$ by $- 4$ .

$- f (x) = - 3 x + 4 - (2 x^{2})$

Step 3.1.3.3

Multiply $2$ by $- 1$ .

$- f (x) = - 3 x + 4 - 2 x^{2}$

Step 3.2

Since $- 3 x - 4 + 2 x^{2}$ $\neq$ $- 3 x + 4 - 2 x^{2}$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5

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