Precalculus Examples

$f (x) = | x + 1 | + | x - 1 |$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = | (- x) + 1 | + | (- x) - 1 |$

Step 1.2

Remove parentheses.

$f (- x) = | - x + 1 | + | - x - 1 |$

Step 2

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

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Step 2.1

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

Step 2.2

Since $| - x + 1 | + | - x - 1 |$ $\neq$ $| x + 1 | + | x - 1 |$ , 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 $| x + 1 | + | x - 1 |$ by $- 1$ .

$- f (x) = - (| x + 1 | + | x - 1 |)$

Step 3.1.2

Apply the distributive property.

$- f (x) = - | x + 1 | - | x - 1 |$

Step 3.2

Since $| - x + 1 | + | - x - 1 |$ $\neq$ $- | x + 1 | - | x - 1 |$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5