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Examples

f (x) = 7 x^{2} + 5 x - 4

$f (x) = 7 x^{2} + 5 x - 4$

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) = 7 {(- x)}^{2} + 5 (- x) - 4

$f (- x) = 7 {(- x)}^{2} + 5 (- x) - 4$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to

- x

$- x$ .

f (- x) = 7 ({(- 1)}^{2} x^{2}) + 5 (- x) - 4

$f (- x) = 7 ({(- 1)}^{2} x^{2}) + 5 (- x) - 4$

Step 1.2.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

f (- x) = 7 (1 x^{2}) + 5 (- x) - 4

$f (- x) = 7 (1 x^{2}) + 5 (- x) - 4$

Step 1.2.3

Multiply

x^{2}

$x^{2}$ by

1

$1$ .

f (- x) = 7 x^{2} + 5 (- x) - 4

$f (- x) = 7 x^{2} + 5 (- x) - 4$

Step 1.2.4

Multiply

- 1

$- 1$ by

5

$5$ .

f (- x) = 7 x^{2} - 5 x - 4

$f (- x) = 7 x^{2} - 5 x - 4$

f (- x) = 7 x^{2} - 5 x - 4

$f (- x) = 7 x^{2} - 5 x - 4$

f (- x) = 7 x^{2} - 5 x - 4

$f (- x) = 7 x^{2} - 5 x - 4$

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

7 x^{2} - 5 x - 4

$7 x^{2} - 5 x - 4$

\neq

$\neq$

7 x^{2} + 5 x - 4

$7 x^{2} + 5 x - 4$ , 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)

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

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Step 3.1

Find

- f (x)

$- f (x)$ .

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Step 3.1.1

Multiply

7 x^{2} + 5 x - 4

$7 x^{2} + 5 x - 4$ by

- 1

$- 1$ .

- f (x) = - (7 x^{2} + 5 x - 4)

$- f (x) = - (7 x^{2} + 5 x - 4)$

Step 3.1.2

Apply the distributive property.

- f (x) = - (7 x^{2}) - (5 x) + 4

$- f (x) = - (7 x^{2}) - (5 x) + 4$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply

7

$7$ by

- 1

$- 1$ .

- f (x) = - 7 x^{2} - (5 x) + 4

$- f (x) = - 7 x^{2} - (5 x) + 4$

Step 3.1.3.2

Multiply

5

$5$ by

- 1

$- 1$ .

- f (x) = - 7 x^{2} - 5 x + 4

$- f (x) = - 7 x^{2} - 5 x + 4$

Step 3.1.3.3

Multiply

- 1

$- 1$ by

- 4

$- 4$ .

- f (x) = - 7 x^{2} - 5 x + 4

$- f (x) = - 7 x^{2} - 5 x + 4$

- f (x) = - 7 x^{2} - 5 x + 4

$- f (x) = - 7 x^{2} - 5 x + 4$

- f (x) = - 7 x^{2} - 5 x + 4

$- f (x) = - 7 x^{2} - 5 x + 4$

Step 3.2

Since

7 x^{2} - 5 x - 4

$7 x^{2} - 5 x - 4$

\neq

$\neq$

- 7 x^{2} - 5 x + 4

$- 7 x^{2} - 5 x + 4$ , 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{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

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