Precalculus Examples

$f (x) = \frac{1}{x + 3}$ , $g (x) = \frac{- 2}{x}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

$f (\frac{- 2}{x}) = \frac{1}{(\frac{- 2}{x}) + 3}$

Step 3

Simplify the denominator.

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Step 3.1

Move the negative in front of the fraction.

$f (\frac{- 2}{x}) = \frac{1}{- \frac{2}{x} + 3}$

Step 3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$f (\frac{- 2}{x}) = \frac{1}{- \frac{2}{x} + \frac{3 x}{x}}$

Step 3.3

Combine the numerators over the common denominator.

$f (\frac{- 2}{x}) = \frac{1}{\frac{- 2 + 3 x}{x}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{- 2}{x}) = 1 (\frac{x}{- 2 + 3 x})$

Step 5

Multiply $\frac{x}{- 2 + 3 x}$ by $1$ .

$f (\frac{- 2}{x}) = \frac{x}{- 2 + 3 x}$

Step 6

Rewrite $- 2$ as $- 1 (2)$ .

$f (\frac{- 2}{x}) = \frac{x}{- 1 \cdot 2 + 3 x}$

Step 7

Factor $- 1$ out of $3 x$ .

$f (\frac{- 2}{x}) = \frac{x}{- 1 \cdot 2 - (- 3 x)}$

Step 8

Factor $- 1$ out of $- 1 (2) - (- 3 x)$ .

$f (\frac{- 2}{x}) = \frac{x}{- 1 (2 - 3 x)}$

Step 9

Move the negative in front of the fraction.

$f (\frac{- 2}{x}) = - \frac{x}{2 - 3 x}$