Find f(g(x)) f(x)=(6+x)/(x-4) , g(x)=3/x

$f (x) = \frac{6 + x}{x - 4}$ , $g (x) = \frac{3}{x}$

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 $f$ .

$f (\frac{3}{x}) = \frac{6 + \frac{3}{x}}{(\frac{3}{x}) - 4}$

Step 3

Remove parentheses.

$f (\frac{3}{x}) = \frac{6 + \frac{3}{x}}{(\frac{3}{x}) - 4}$

Step 4

Multiply the numerator and denominator of the complex fraction by $x$ .

Multiply $\frac{6 + \frac{3}{x}}{\frac{3}{x} - 4}$ by $\frac{x}{x}$ .

$f (\frac{3}{x}) = \frac{x}{x} \cdot \frac{6 + \frac{3}{x}}{\frac{3}{x} - 4}$

Combine.

$f (\frac{3}{x}) = \frac{x (6 + \frac{3}{x})}{x (\frac{3}{x} - 4)}$

Step 5

Apply the distributive property.

$f (\frac{3}{x}) = \frac{x \cdot 6 + x (\frac{3}{x})}{x (\frac{3}{x}) + x \cdot - 4}$

Step 6

Simplify by cancelling.

Cancel the common factor of $x$ .

Cancel the common factor.

$f (\frac{3}{x}) = \frac{x \cdot 6 + x (\frac{3}{x})}{x (\frac{3}{x}) + x \cdot - 4}$

Rewrite the expression.

$f (\frac{3}{x}) = \frac{x \cdot 6 + 3}{x (\frac{3}{x}) + x \cdot - 4}$

Cancel the common factor of $x$ .

Cancel the common factor.

$f (\frac{3}{x}) = \frac{x \cdot 6 + 3}{x (\frac{3}{x}) + x \cdot - 4}$

Rewrite the expression.

$f (\frac{3}{x}) = \frac{x \cdot 6 + 3}{3 + x \cdot - 4}$

Step 7

Simplify the numerator.

Factor $3$ out of $x \cdot 6 + 3$ .

Factor $3$ out of $x \cdot 6$ .

$f (\frac{3}{x}) = \frac{3 (x \cdot 2) + 3}{3 + x \cdot - 4}$

Factor $3$ out of $3$ .

$f (\frac{3}{x}) = \frac{3 (x \cdot 2) + 3 (1)}{3 + x \cdot - 4}$

Factor $3$ out of $3 (x \cdot 2) + 3 (1)$ .

$f (\frac{3}{x}) = \frac{3 (x \cdot 2 + 1)}{3 + x \cdot - 4}$

Move $2$ to the left of $x$ .

$f (\frac{3}{x}) = \frac{3 (2 x + 1)}{3 + x \cdot - 4}$

Step 8

Move $- 4$ to the left of $x$ .

$f (\frac{3}{x}) = \frac{3 (2 x + 1)}{3 - 4 x}$