Find f(g(x)) f(x)=3x-2 , g(x)=9/(x+2)

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{9}{x + 2})$ by substituting in the value of $g$ into $f$ .

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

Step 3

Multiply $3 (\frac{9}{x + 2})$ .

Combine $3$ and $\frac{9}{x + 2}$ .

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

Multiply $3$ by $9$ .

$f (\frac{9}{x + 2}) = \frac{27}{x + 2} - 2$

Step 4

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$f (\frac{9}{x + 2}) = \frac{27}{x + 2} - 2 \cdot \frac{x + 2}{x + 2}$

Step 5

Simplify terms.

Combine $- 2$ and $\frac{x + 2}{x + 2}$ .

$f (\frac{9}{x + 2}) = \frac{27}{x + 2} + \frac{- 2 (x + 2)}{x + 2}$

Combine the numerators over the common denominator.

$f (\frac{9}{x + 2}) = \frac{27 - 2 (x + 2)}{x + 2}$

Step 6

Simplify the numerator.

Apply the distributive property.

$f (\frac{9}{x + 2}) = \frac{27 - 2 x - 2 \cdot 2}{x + 2}$

Multiply $- 2$ by $2$ .

$f (\frac{9}{x + 2}) = \frac{27 - 2 x - 4}{x + 2}$

Subtract $4$ from $27$ .

$f (\frac{9}{x + 2}) = \frac{- 2 x + 23}{x + 2}$

Step 7

Simplify with factoring out.

Factor $- 1$ out of $- 2 x$ .

$f (\frac{9}{x + 2}) = \frac{- (2 x) + 23}{x + 2}$

Rewrite $23$ as $- 1 (- 23)$ .

$f (\frac{9}{x + 2}) = \frac{- (2 x) - 1 \cdot - 23}{x + 2}$

Factor $- 1$ out of $- (2 x) - 1 (- 23)$ .

$f (\frac{9}{x + 2}) = \frac{- (2 x - 23)}{x + 2}$

Simplify the expression.

Rewrite $- (2 x - 23)$ as $- 1 (2 x - 23)$ .

$f (\frac{9}{x + 2}) = \frac{- 1 (2 x - 23)}{x + 2}$

Move the negative in front of the fraction.

$f (\frac{9}{x + 2}) = - \frac{2 x - 23}{x + 2}$