Algebra Examples

$f (x) = \frac{(x + 16)}{x}$ ; $g (x) = \frac{(2 x)}{(x + 4)}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{x + 16}{x})$ by substituting in the value of $f$ into $g$ .

$g (\frac{x + 16}{x}) = \frac{2 (\frac{x + 16}{x})}{(\frac{x + 16}{x}) + 4}$

Step 3

Combine $2$ and $\frac{x + 16}{x}$ .

$g (\frac{x + 16}{x}) = \frac{\frac{2 (x + 16)}{x}}{\frac{x + 16}{x} + 4}$

Step 4

Simplify the denominator.

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Step 4.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$g (\frac{x + 16}{x}) = \frac{\frac{2 (x + 16)}{x}}{\frac{x + 16}{x} + \frac{4 x}{x}}$

Step 4.2

Combine the numerators over the common denominator.

$g (\frac{x + 16}{x}) = \frac{\frac{2 (x + 16)}{x}}{\frac{x + 16 + 4 x}{x}}$

Step 4.3

Reorder terms.

$g (\frac{x + 16}{x}) = \frac{\frac{2 (x + 16)}{x}}{\frac{4 x + x + 16}{x}}$

Step 4.4

Add $4 x$ and $x$ .

$g (\frac{x + 16}{x}) = \frac{\frac{2 (x + 16)}{x}}{\frac{5 x + 16}{x}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{x + 16}{x}) = \frac{2 (x + 16)}{x} \cdot \frac{x}{5 x + 16}$

Step 6

Cancel the common factor of $x$ .

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Step 6.1

Cancel the common factor.

$g (\frac{x + 16}{x}) = \frac{2 (x + 16)}{x} \cdot \frac{x}{5 x + 16}$

Step 6.2

Rewrite the expression.

$g (\frac{x + 16}{x}) = 2 (x + 16) (\frac{1}{5 x + 16})$

Step 7

Combine $\frac{1}{5 x + 16}$ and $2$ .

$g (\frac{x + 16}{x}) = \frac{2}{5 x + 16} \cdot (x + 16)$

Step 8

Multiply $\frac{2}{5 x + 16}$ by $x + 16$ .

$g (\frac{x + 16}{x}) = \frac{2 (x + 16)}{5 x + 16}$