Find f(g(x)) f(x)=(x+16)/x , g(x)=(2x)/(x+4)

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{2 x}{x + 4})$ by substituting in the value of $g$ into $f$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

To write $16$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Factor $2$ out of $2 x + 16 (x + 4)$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $16 (x + 4)$ .

$f (\frac{2 x}{x + 4}) = \frac{2 (x) + 2 (8 (x + 4))}{x + 4} \cdot \frac{x + 4}{2 x}$

Factor $2$ out of $2 (x) + 2 (8 (x + 4))$ .

$f (\frac{2 x}{x + 4}) = \frac{2 (x + 8 (x + 4))}{x + 4} \cdot \frac{x + 4}{2 x}$

Apply the distributive property.

$f (\frac{2 x}{x + 4}) = \frac{2 (x + 8 x + 8 \cdot 4)}{x + 4} \cdot \frac{x + 4}{2 x}$

Multiply $8$ by $4$ .

$f (\frac{2 x}{x + 4}) = \frac{2 (x + 8 x + 32)}{x + 4} \cdot \frac{x + 4}{2 x}$

Add $x$ and $8 x$ .

$f (\frac{2 x}{x + 4}) = \frac{2 (9 x + 32)}{x + 4} \cdot \frac{x + 4}{2 x}$

Step 7

Simplify terms.

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

$f (\frac{2 x}{x + 4}) = \frac{2 (9 x + 32)}{x + 4} \cdot \frac{x + 4}{2 (x)}$

Cancel the common factor.

$f (\frac{2 x}{x + 4}) = \frac{2 (9 x + 32)}{x + 4} \cdot \frac{x + 4}{2 x}$

Rewrite the expression.

$f (\frac{2 x}{x + 4}) = \frac{9 x + 32}{x + 4} \cdot \frac{x + 4}{x}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$f (\frac{2 x}{x + 4}) = \frac{9 x + 32}{x + 4} \cdot \frac{x + 4}{x}$

Rewrite the expression.

$f (\frac{2 x}{x + 4}) = (9 x + 32) (\frac{1}{x})$

Multiply $9 x + 32$ by $\frac{1}{x}$ .

$f (\frac{2 x}{x + 4}) = \frac{9 x + 32}{x}$

Find f(g(x)) f(x)=(x+16)/x , g(x)=(2x)/(x+4)

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