Precalculus Examples

$y = f (x) - g (x)$

Step 1

Write $y = f (x) - g (x)$ as a function.

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

Step 2

Find $f (- x)$ .

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Step 2.1

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite using the commutative property of multiplication.

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

Step 2.2.2

Rewrite using the commutative property of multiplication.

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

Step 2.2.3

Multiply $- 1$ by $- 1$ .

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

Step 2.2.4

Multiply $g$ by $1$ .

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

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 $- f x + g x$ $\neq$ $f (x) - g x$ , 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 $f (x) - g x$ by $- 1$ .

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Multiply $- (- g x)$ .

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.3.2

Multiply $g$ by $1$ .

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

Step 4.2

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

The function is not odd

Step 5

The function is neither odd nor even

Step 6