Find f(g(x)) f(x)=9/(x-6) , g(x)=9/x+6

$f (x) = \frac{9}{x - 6}$ , $g (x) = \frac{9}{x} + 6$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{9}{x} + 6)$ by substituting in the value of $g$ into $f$ .

$f (\frac{9}{x} + 6) = \frac{9}{(\frac{9}{x} + 6) - 6}$

Step 3

Cancel the common factor of $9$ and $(\frac{9}{x} + 6) - 6$ .

Factor $3$ out of $9$ .

$f (\frac{9}{x} + 6) = \frac{3 (3)}{(\frac{9}{x} + 6) - 6}$

Cancel the common factors.

Factor $3$ out of $\frac{9}{x}$ .

$f (\frac{9}{x} + 6) = \frac{3 \cdot 3}{3 (\frac{3}{x}) + 6 - 6}$

Factor $3$ out of $6$ .

$f (\frac{9}{x} + 6) = \frac{3 (3)}{3 (\frac{3}{x}) + 3 \cdot 2 - 6}$

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

$f (\frac{9}{x} + 6) = \frac{3 (3)}{3 (\frac{3}{x} + 2) - 6}$

Factor $3$ out of $- 6$ .

$f (\frac{9}{x} + 6) = \frac{3 \cdot 3}{3 (\frac{3}{x} + 2) + 3 \cdot - 2}$

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

$f (\frac{9}{x} + 6) = \frac{3 (3)}{3 (\frac{3}{x} + 2 - 2)}$

Cancel the common factor.

$f (\frac{9}{x} + 6) = \frac{3 \cdot 3}{3 (\frac{3}{x} + 2 - 2)}$

Rewrite the expression.

$f (\frac{9}{x} + 6) = \frac{3}{\frac{3}{x} + 2 - 2}$

Step 4

Simplify the denominator.

Subtract $2$ from $2$ .

$f (\frac{9}{x} + 6) = \frac{3}{\frac{3}{x} + 0}$

Add $\frac{3}{x}$ and $0$ .

$f (\frac{9}{x} + 6) = \frac{3}{\frac{3}{x}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{9}{x} + 6) = 3 (\frac{x}{3})$

Step 6

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{9}{x} + 6) = 3 (\frac{x}{3})$

Rewrite the expression.

$f (\frac{9}{x} + 6) = x$