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

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{6}{x} + 3)$ by substituting in the value of $g$ into $f$ .

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

Step 3

Cancel the common factor of $6$ and $(\frac{6}{x} + 3) - 3$ .

Factor $3$ out of $6$ .

$f (\frac{6}{x} + 3) = \frac{3 (2)}{(\frac{6}{x} + 3) - 3}$

Cancel the common factors.

Factor $3$ out of $\frac{6}{x}$ .

$f (\frac{6}{x} + 3) = \frac{3 \cdot 2}{3 (\frac{2}{x}) + 3 - 3}$

Factor $3$ out of $3$ .

$f (\frac{6}{x} + 3) = \frac{3 (2)}{3 (\frac{2}{x}) + 3 \cdot 1 - 3}$

Factor $3$ out of $3 \frac{2}{x} + 3 \cdot 1$ .

$f (\frac{6}{x} + 3) = \frac{3 (2)}{3 (\frac{2}{x} + 1) - 3}$

Factor $3$ out of $- 3$ .

$f (\frac{6}{x} + 3) = \frac{3 \cdot 2}{3 (\frac{2}{x} + 1) + 3 \cdot - 1}$

Factor $3$ out of $3 (\frac{2}{x} + 1) + 3 (- 1)$ .

$f (\frac{6}{x} + 3) = \frac{3 (2)}{3 (\frac{2}{x} + 1 - 1)}$

Cancel the common factor.

$f (\frac{6}{x} + 3) = \frac{3 \cdot 2}{3 (\frac{2}{x} + 1 - 1)}$

Rewrite the expression.

$f (\frac{6}{x} + 3) = \frac{2}{\frac{2}{x} + 1 - 1}$

Step 4

Simplify the denominator.

Subtract $1$ from $1$ .

$f (\frac{6}{x} + 3) = \frac{2}{\frac{2}{x} + 0}$

Add $\frac{2}{x}$ and $0$ .

$f (\frac{6}{x} + 3) = \frac{2}{\frac{2}{x}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{6}{x} + 3) = 2 (\frac{x}{2})$

Step 6

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{6}{x} + 3) = 2 (\frac{x}{2})$

Rewrite the expression.

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