Calculus Examples

$f (x) = 3 x^{2} - 3$ ; $g (x) = \frac{8}{x}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (3 x^{2} - 3)$ by substituting in the value of $f$ into $g$ .

$g (3 x^{2} - 3) = \frac{8}{3 x^{2} - 3}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $3$ out of $3 x^{2} - 3$ .

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Step 3.1.1

Factor $3$ out of $3 x^{2}$ .

$g (3 x^{2} - 3) = \frac{8}{3 (x^{2}) - 3}$

Step 3.1.2

Factor $3$ out of $- 3$ .

$g (3 x^{2} - 3) = \frac{8}{3 (x^{2}) + 3 (- 1)}$

Step 3.1.3

Factor $3$ out of $3 (x^{2}) + 3 (- 1)$ .

$g (3 x^{2} - 3) = \frac{8}{3 (x^{2} - 1)}$

Step 3.2

Rewrite $1$ as $1^{2}$ .

$g (3 x^{2} - 3) = \frac{8}{3 (x^{2} - 1^{2})}$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$g (3 x^{2} - 3) = \frac{8}{3 (x + 1) (x - 1)}$