Algebra Examples

$y = \sin (x) + \cos (x)$

Step 1

Write $y = \sin (x) + \cos (x)$ as a function.

$f (x) = \sin (x) + \cos (x)$

Step 2

Find $f (- x)$ .

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Step 2.1

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

$f (- x) = \sin (- x) + \cos (- x)$

Step 2.2

Simplify each term.

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Step 2.2.1

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

$f (- x) = - \sin (x) + \cos (- x)$

Step 2.2.2

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

$f (- x) = - \sin (x) + \cos (x)$

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 $- \sin (x) + \cos (x)$ $\neq$ $\sin (x) + \cos (x)$ , 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

Find $- f (x)$ .

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Step 4.1.1

Multiply $\sin (x) + \cos (x)$ by $- 1$ .

$- f (x) = - (\sin (x) + \cos (x))$

Step 4.1.2

Apply the distributive property.

$- f (x) = - \sin (x) - \cos (x)$

Step 4.2

Since $- \sin (x) + \cos (x)$ $\neq$ $- \sin (x) - \cos (x)$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6