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

Reorder factors in $f (x) g x$ .

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

Step 3

Find $f (- x)$ .

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Step 3.1

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

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

Step 3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.1

Move $x$ .

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

Step 3.2.2

Multiply $x$ by $x$ .

$f (- x) = g (- x^{2}) \cdot (f \cdot (- 1))$

Step 3.3

Rewrite using the commutative property of multiplication.

$f (- x) = - g x^{2} \cdot (f \cdot (- 1))$

Step 3.4

Move $- 1$ to the left of $f$ .

$f (- x) = - g x^{2} \cdot (- 1 f)$

Step 3.5

Rewrite $- 1 f$ as $- f$ .

$f (- x) = - g x^{2} \cdot (- f)$

Step 3.6

Multiply $- g x^{2} (- f)$ .

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Step 3.6.1

Multiply $- 1$ by $- 1$ .

$f (- x) = 1 g x^{2} f$

Step 3.6.2

Multiply $g$ by $1$ .

$f (- x) = g x^{2} f$

Step 4

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

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Step 4.1

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

Step 4.2

Since $g x^{2} f = g x f (x)$ , the function is even.

The function is even

Step 5