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Precalculus Examples

f (x) = \csc (x)

$f (x) = \csc (x)$

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) = \csc (- x)

$f (- x) = \csc (- x)$

Step 1.2

Since

\csc (- x)

$\csc (- x)$ is an odd function, rewrite

\csc (- x)

$\csc (- x)$ as

- \csc (x)

$- \csc (x)$ .

f (- x) = - \csc (x)

$f (- x) = - \csc (x)$

f (- x) = - \csc (x)

$f (- x) = - \csc (x)$

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

- \csc (x)

$- \csc (x)$

\neq

$\neq$

\csc (x)

$\csc (x)$ , 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

Multiply

- 1

$- 1$ by

\csc (x)

$\csc (x)$ .

- f (x) = - \csc (x)

$- f (x) = - \csc (x)$

Step 3.2

Since

- \csc (x) = - \csc (x)

$- \csc (x) = - \csc (x)$ , the function is odd.

The function is odd

The function is odd

Step 4

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

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