Find f(g(x)) f(x)=(x-9)/(x+8) g(x)=(x+3)/(x-8)

$f (x) = \frac{x - 9}{x + 8}$ $g (x) = \frac{x + 3}{x - 8}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{x + 3}{x - 8})$ by substituting in the value of $g$ into $f$ .

$f (\frac{x + 3}{x - 8}) = \frac{(\frac{x + 3}{x - 8}) - 9}{(\frac{x + 3}{x - 8}) + 8}$

Step 3

Multiply the numerator and denominator of the complex fraction by $x - 8$ .

Multiply $\frac{\frac{x + 3}{x - 8} - 9}{\frac{x + 3}{x - 8} + 8}$ by $\frac{x - 8}{x - 8}$ .

$f (\frac{x + 3}{x - 8}) = \frac{x - 8}{x - 8} \cdot \frac{\frac{x + 3}{x - 8} - 9}{\frac{x + 3}{x - 8} + 8}$

Combine.

$f (\frac{x + 3}{x - 8}) = \frac{(x - 8) (\frac{x + 3}{x - 8} - 9)}{(x - 8) (\frac{x + 3}{x - 8} + 8)}$

Step 4

Apply the distributive property.

$f (\frac{x + 3}{x - 8}) = \frac{(x - 8) (\frac{x + 3}{x - 8}) + (x - 8) \cdot - 9}{(x - 8) (\frac{x + 3}{x - 8}) + (x - 8) \cdot 8}$

Step 5

Simplify by cancelling.

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$f (\frac{x + 3}{x - 8}) = \frac{(x - 8) (\frac{x + 3}{x - 8}) + (x - 8) \cdot - 9}{(x - 8) (\frac{x + 3}{x - 8}) + (x - 8) \cdot 8}$

Rewrite the expression.

$f (\frac{x + 3}{x - 8}) = \frac{x + 3 + (x - 8) \cdot - 9}{(x - 8) (\frac{x + 3}{x - 8}) + (x - 8) \cdot 8}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$f (\frac{x + 3}{x - 8}) = \frac{x + 3 + (x - 8) \cdot - 9}{(x - 8) (\frac{x + 3}{x - 8}) + (x - 8) \cdot 8}$

Rewrite the expression.

$f (\frac{x + 3}{x - 8}) = \frac{x + 3 + (x - 8) \cdot - 9}{x + 3 + (x - 8) \cdot 8}$

Step 6

Simplify the numerator.

Apply the distributive property.

$f (\frac{x + 3}{x - 8}) = \frac{x + 3 + x \cdot - 9 - 8 \cdot - 9}{x + 3 + (x - 8) \cdot 8}$

Move $- 9$ to the left of $x$ .

$f (\frac{x + 3}{x - 8}) = \frac{x + 3 - 9 \cdot x - 8 \cdot - 9}{x + 3 + (x - 8) \cdot 8}$

Multiply $- 8$ by $- 9$ .

$f (\frac{x + 3}{x - 8}) = \frac{x + 3 - 9 \cdot x + 72}{x + 3 + (x - 8) \cdot 8}$

Subtract $9 x$ from $x$ .

$f (\frac{x + 3}{x - 8}) = \frac{- 8 x + 3 + 72}{x + 3 + (x - 8) \cdot 8}$

Add $3$ and $72$ .

$f (\frac{x + 3}{x - 8}) = \frac{- 8 x + 75}{x + 3 + (x - 8) \cdot 8}$

Step 7

Simplify the denominator.

Apply the distributive property.

$f (\frac{x + 3}{x - 8}) = \frac{- 8 x + 75}{x + 3 + x \cdot 8 - 8 \cdot 8}$

Move $8$ to the left of $x$ .

$f (\frac{x + 3}{x - 8}) = \frac{- 8 x + 75}{x + 3 + 8 \cdot x - 8 \cdot 8}$

Multiply $- 8$ by $8$ .

$f (\frac{x + 3}{x - 8}) = \frac{- 8 x + 75}{x + 3 + 8 \cdot x - 64}$

Add $x$ and $8 x$ .

$f (\frac{x + 3}{x - 8}) = \frac{- 8 x + 75}{9 x + 3 - 64}$

Subtract $64$ from $3$ .

$f (\frac{x + 3}{x - 8}) = \frac{- 8 x + 75}{9 x - 61}$

Step 8

Simplify with factoring out.

Factor $- 1$ out of $- 8 x$ .

$f (\frac{x + 3}{x - 8}) = \frac{- (8 x) + 75}{9 x - 61}$

Rewrite $75$ as $- 1 (- 75)$ .

$f (\frac{x + 3}{x - 8}) = \frac{- (8 x) - 1 \cdot - 75}{9 x - 61}$

Factor $- 1$ out of $- (8 x) - 1 (- 75)$ .

$f (\frac{x + 3}{x - 8}) = \frac{- (8 x - 75)}{9 x - 61}$

Simplify the expression.

Rewrite $- (8 x - 75)$ as $- 1 (8 x - 75)$ .

$f (\frac{x + 3}{x - 8}) = \frac{- 1 (8 x - 75)}{9 x - 61}$

Move the negative in front of the fraction.

$f (\frac{x + 3}{x - 8}) = - \frac{8 x - 75}{9 x - 61}$

Find f(g(x)) f(x)=(x-9)/(x+8) g(x)=(x+3)/(x-8)

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