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Examples

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

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

Step 1

Find

f (- x)

$f (- x)$ .

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Step 1.1

Find

f (- x)

$f (- x)$ by substituting

- x

$- x$ for all occurrence of

x

$x$ in

f (x)

$f (x)$ .

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

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

Step 1.2

Simplify each term.

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Step 1.2.1

Multiply

- 1

$- 1$ by

3

$3$ .

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

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

Step 1.2.2

Apply the product rule to

- x

$- x$ .

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

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

Step 1.2.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 1.2.4

Multiply

x^{2}

$x^{2}$ by

1

$1$ .

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

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

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

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

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

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

Step 2

A function is even if

f (- x) = f (x)

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

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Step 2.1

Check if

f (- x) = f (x)

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

Step 2.2

Since

- 3 x - 4 + 2 x^{2}

$- 3 x - 4 + 2 x^{2}$

\neq

$\neq$

3 x - 4 + 2 x^{2}

$3 x - 4 + 2 x^{2}$ , the function is not even.

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

$- 1$ .

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

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

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

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

The function is not odd

Step 4

The function is neither odd nor even

Step 5

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$