Find f(g(x)) f(x)=(8x)/(x+4) , g(x)=3/x

$f (x) = \frac{8 x}{x + 4}$ , $g (x) = \frac{3}{x}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{3}{x})$ by substituting in the value of $g$ into $f$ .

$f (\frac{3}{x}) = \frac{8 (\frac{3}{x})}{(\frac{3}{x}) + 4}$

Step 3

Combine $8$ and $\frac{3}{x}$ .

$f (\frac{3}{x}) = \frac{\frac{8 \cdot 3}{x}}{\frac{3}{x} + 4}$

Step 4

Simplify the denominator.

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

$f (\frac{3}{x}) = \frac{\frac{8 \cdot 3}{x}}{\frac{3}{x} + \frac{4 x}{x}}$

Combine the numerators over the common denominator.

$f (\frac{3}{x}) = \frac{\frac{8 \cdot 3}{x}}{\frac{3 + 4 x}{x}}$

Step 5

Multiply $8$ by $3$ .

$f (\frac{3}{x}) = \frac{\frac{24}{x}}{\frac{3 + 4 x}{x}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{3}{x}) = \frac{24}{x} \cdot \frac{x}{3 + 4 x}$

Step 7

Cancel the common factor of $x$ .

Cancel the common factor.

$f (\frac{3}{x}) = \frac{24}{x} \cdot \frac{x}{3 + 4 x}$

Rewrite the expression.

$f (\frac{3}{x}) = 24 (\frac{1}{3 + 4 x})$

Step 8

Combine $24$ and $\frac{1}{3 + 4 x}$ .

$f (\frac{3}{x}) = \frac{24}{3 + 4 x}$