Precalculus Examples

$\csc (x)$

Step 1

Write $\csc (x)$ as a function.

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

Step 2

Find $f (- x)$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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Step 3.1

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

Step 3.2

Since $- \csc (x)$ $\neq$ $\csc (x)$ , the function is not even.

The function is not even

Step 4

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

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Step 4.1

Multiply $- 1$ by $\csc (x)$ .

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

Step 4.2

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

The function is odd

Step 5