Precalculus Examples

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

Step 1

Write $f (- x) + f (x) = 0$ as a function.

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

Step 2

Add $f (x)$ to both sides of the equation.

$f (- x) + f (x) = 0$

Step 3

Rewrite using the commutative property of multiplication.

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

Step 4

Find $f (- x)$ .

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Step 4.1

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

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

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite using the commutative property of multiplication.

$f (- x) = - 1 \cdot (- 1 f x) + f (- x)$

Step 4.2.2

Multiply $- 1$ by $- 1$ .

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

Step 4.2.3

Multiply $f$ by $1$ .

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

Step 4.2.4

Rewrite using the commutative property of multiplication.

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

Step 4.3

Subtract $f x$ from $f x$ .

$f (- x) = 0$

Step 5

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

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Step 5.1

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

Step 5.2

Since $0$ $\neq$ $- f x + f (x)$ , the function is not even.

The function is not even

Step 6

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

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Step 6.1

Find $- f (x)$ .

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Step 6.1.1

Multiply $- f x + f (x)$ by $- 1$ .

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Multiply $- (- f x)$ .

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Step 6.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.3.2

Multiply $f$ by $1$ .

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

Step 6.2

Since $0$ $\neq$ $f x - f (x)$ , the function is not odd.

The function is not odd

Step 7

The function is neither odd nor even

Step 8