Algebra Examples

$f (x) = \frac{x^{2} - 36}{x + 6}$ , $g (x) = x - 6$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x - 6)$ by substituting in the value of $g$ into $f$ .

$f (x - 6) = \frac{{(x - 6)}^{2} - 36}{(x - 6) + 6}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $36$ as $6^{2}$ .

$f (x - 6) = \frac{{(x - 6)}^{2} - 6^{2}}{x - 6 + 6}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x - 6$ and $b = 6$ .

$f (x - 6) = \frac{(x - 6 + 6) (x - 6 - 6)}{x - 6 + 6}$

Step 3.3

Simplify.

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Step 3.3.1

Add $- 6$ and $6$ .

$f (x - 6) = \frac{(x + 0) (x - 6 - 6)}{x - 6 + 6}$

Step 3.3.2

Add $x$ and $0$ .

$f (x - 6) = \frac{x (x - 6 - 6)}{x - 6 + 6}$

Step 3.3.3

Subtract $6$ from $- 6$ .

$f (x - 6) = \frac{x (x - 12)}{x - 6 + 6}$

Step 4

Simplify the denominator.

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Step 4.1

Add $- 6$ and $6$ .

$f (x - 6) = \frac{x (x - 12)}{x + 0}$

Step 4.2

Add $x$ and $0$ .

$f (x - 6) = \frac{x (x - 12)}{x}$

Step 5

Cancel the common factor of $x$ .

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Step 5.1

Cancel the common factor.

$f (x - 6) = \frac{x (x - 12)}{x}$

Step 5.2

Divide $x - 12$ by $1$ .

$f (x - 6) = x - 12$