Find f(g(x)) f(x)=(x^2-8x+16)/(x+2x) , g(x)=(6x)/(x-4)

$f (x) = \frac{x^{2} - 8 x + 16}{x + 2 x}$ , $g (x) = \frac{(6 x)}{(x - 4)}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{6 x}{x - 4})$ by substituting in the value of $g$ into $f$ .

$f (\frac{6 x}{x - 4}) = \frac{{(\frac{6 x}{x - 4})}^{2} - 8 \frac{6 x}{x - 4} + 16}{(\frac{6 x}{x - 4}) + 2 (\frac{6 x}{x - 4})}$

Step 3

Multiply the numerator and denominator of the complex fraction by $x - 4$ .

Multiply $\frac{{(\frac{6 x}{x - 4})}^{2} - 8 \frac{6 x}{x - 4} + 16}{\frac{6 x}{x - 4} + 2 \frac{6 x}{x - 4}}$ by $\frac{x - 4}{x - 4}$ .

$f (\frac{6 x}{x - 4}) = \frac{x - 4}{x - 4} \cdot \frac{{(\frac{6 x}{x - 4})}^{2} - 8 \frac{6 x}{x - 4} + 16}{\frac{6 x}{x - 4} + 2 (\frac{6 x}{x - 4})}$

Combine.

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) ({(\frac{6 x}{x - 4})}^{2} - 8 \frac{6 x}{x - 4} + 16)}{(x - 4) (\frac{6 x}{x - 4} + 2 (\frac{6 x}{x - 4}))}$

Step 4

Apply the distributive property.

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{6 x}{x - 4})}^{2} + (x - 4) (- 8 \frac{6 x}{x - 4}) + (x - 4) \cdot 16}{(x - 4) (\frac{6 x}{x - 4}) + (x - 4) (2 (\frac{6 x}{x - 4}))}$

Step 5

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{6 x}{x - 4})}^{2} + (x - 4) (- 8 \frac{6 x}{x - 4}) + (x - 4) \cdot 16}{(x - 4) (\frac{6 x}{x - 4}) + (x - 4) (2 (\frac{6 x}{x - 4}))}$

Rewrite the expression.

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{6 x}{x - 4})}^{2} + (x - 4) (- 8 \frac{6 x}{x - 4}) + (x - 4) \cdot 16}{6 x + (x - 4) (2 (\frac{6 x}{x - 4}))}$

Step 6

Simplify the numerator.

Factor $x - 4$ out of $(x - 4) {(\frac{6 x}{x - 4})}^{2} + (x - 4) (- 8 \frac{6 x}{x - 4}) + (x - 4) \cdot 16$ .

Factor $x - 4$ out of $(x - 4) {(\frac{6 x}{x - 4})}^{2}$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) ({(\frac{6 x}{x - 4})}^{2}) + (x - 4) (- 8 \frac{6 x}{x - 4}) + (x - 4) \cdot 16}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Factor $x - 4$ out of $(x - 4) \cdot 16$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) ({(\frac{6 x}{x - 4})}^{2}) + (x - 4) (- 8 \frac{6 x}{x - 4}) + (x - 4) (16)}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Factor $x - 4$ out of $(x - 4) ({(\frac{6 x}{x - 4})}^{2}) + (x - 4) (- 8 \frac{6 x}{x - 4})$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) ({(\frac{6 x}{x - 4})}^{2} - 8 \frac{6 x}{x - 4}) + (x - 4) (16)}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Factor $x - 4$ out of $(x - 4) ({(\frac{6 x}{x - 4})}^{2} - 8 \frac{6 x}{x - 4}) + (x - 4) (16)$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) ({(\frac{6 x}{x - 4})}^{2} - 8 \frac{6 x}{x - 4} + 16)}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Let $u = \frac{6 x}{x - 4}$ . Substitute $u$ for all occurrences of $\frac{6 x}{x - 4}$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (u^{2} - 8 u + 16)}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (u^{2} - 8 u + 4^{2})}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 u = 2 \cdot u \cdot 4$

Rewrite the polynomial.

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (u^{2} - 2 \cdot u \cdot 4 + 4^{2})}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 4$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(u - 4)}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Replace all occurrences of $u$ with $\frac{6 x}{x - 4}$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{6 x}{x - 4} - 4)}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Simplify.

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{6 x}{x - 4} - 4 \cdot \frac{x - 4}{x - 4})}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Combine $- 4$ and $\frac{x - 4}{x - 4}$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{6 x}{x - 4} + \frac{- 4 (x - 4)}{x - 4})}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Combine the numerators over the common denominator.

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{6 x - 4 (x - 4)}{x - 4})}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Simplify the numerator.

Factor $2$ out of $6 x - 4 (x - 4)$ .

Factor $2$ out of $6 x$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{2 (3 x) - 4 (x - 4)}{x - 4})}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Factor $2$ out of $- 4 (x - 4)$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{2 (3 x) + 2 (- 2 (x - 4))}{x - 4})}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Factor $2$ out of $2 (3 x) + 2 (- 2 (x - 4))$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{2 (3 x - 2 (x - 4))}{x - 4})}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Apply the distributive property.

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{2 (3 x - 2 x - 2 \cdot - 4)}{x - 4})}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Multiply $- 2$ by $- 4$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{2 (3 x - 2 x + 8)}{x - 4})}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Subtract $2 x$ from $3 x$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) {(\frac{2 (x + 8)}{x - 4})}^{2}}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Apply the product rule to $\frac{2 (x + 8)}{x - 4}$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (\frac{{(2 (x + 8))}^{2}}{{(x - 4)}^{2}})}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Apply the product rule to $2 (x + 8)$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (\frac{2^{2} {(x + 8)}^{2}}{{(x - 4)}^{2}})}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Raise $2$ to the power of $2$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (\frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}})}{6 x + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Step 7

Simplify the denominator.

Factor $2$ out of $6 x + (x - 4) \cdot 2 \frac{6 x}{x - 4}$ .

Factor $2$ out of $6 x$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (\frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}})}{2 (3 x) + (x - 4) \cdot (2 (\frac{6 x}{x - 4}))}$

Factor $2$ out of $(x - 4) \cdot 2 \frac{6 x}{x - 4}$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (\frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}})}{2 (3 x) + 2 ((x - 4) (\frac{6 x}{x - 4}))}$

Factor $2$ out of $2 (3 x) + 2 ((x - 4) \frac{6 x}{x - 4})$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (\frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}})}{2 (3 x + (x - 4) (\frac{6 x}{x - 4}))}$

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (\frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}})}{2 (3 x + (x - 4) (\frac{6 x}{x - 4}))}$

Rewrite the expression.

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (\frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}})}{2 (3 x + 6 x)}$

Add $3 x$ and $6 x$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (\frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}})}{2 \cdot (9 x)}$

Multiply $2$ by $9$ .

$f (\frac{6 x}{x - 4}) = \frac{(x - 4) (\frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}})}{18 x}$

Step 8

Simplify terms.

Factor $\frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}}$ out of $\frac{(x - 4) \frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}}}{18 x}$ .

$f (\frac{6 x}{x - 4}) = \frac{4 {(x + 8)}^{2}}{{(x - 4)}^{2}} \cdot \frac{x - 4}{18 x}$

Cancel the common factor of $2$ .

Factor $2$ out of $4 {(x + 8)}^{2}$ .

$f (\frac{6 x}{x - 4}) = \frac{2 (2 {(x + 8)}^{2})}{{(x - 4)}^{2}} \cdot \frac{x - 4}{18 x}$

Factor $2$ out of $18 x$ .

$f (\frac{6 x}{x - 4}) = \frac{2 (2 {(x + 8)}^{2})}{{(x - 4)}^{2}} \cdot \frac{x - 4}{2 (9 x)}$

Cancel the common factor.

$f (\frac{6 x}{x - 4}) = \frac{2 (2 {(x + 8)}^{2})}{{(x - 4)}^{2}} \cdot \frac{x - 4}{2 (9 x)}$

Rewrite the expression.

$f (\frac{6 x}{x - 4}) = \frac{2 {(x + 8)}^{2}}{{(x - 4)}^{2}} \cdot \frac{x - 4}{9 x}$

Cancel the common factor of $x - 4$ .

Factor $x - 4$ out of ${(x - 4)}^{2}$ .

$f (\frac{6 x}{x - 4}) = \frac{2 {(x + 8)}^{2}}{(x - 4) (x - 4)} \cdot \frac{x - 4}{9 x}$

Cancel the common factor.

$f (\frac{6 x}{x - 4}) = \frac{2 {(x + 8)}^{2}}{(x - 4) (x - 4)} \cdot \frac{x - 4}{9 x}$

Rewrite the expression.

$f (\frac{6 x}{x - 4}) = \frac{2 {(x + 8)}^{2}}{x - 4} \cdot \frac{1}{9 x}$

Multiply $\frac{2 {(x + 8)}^{2}}{x - 4}$ by $\frac{1}{9 x}$ .

$f (\frac{6 x}{x - 4}) = \frac{2 {(x + 8)}^{2}}{(x - 4) (9 x)}$

Simplify the expression.

Move $9$ to the left of $x - 4$ .

$f (\frac{6 x}{x - 4}) = \frac{2 {(x + 8)}^{2}}{9 \cdot ((x - 4) x)}$

Reorder factors in $\frac{2 {(x + 8)}^{2}}{9 (x - 4) x}$ .

$f (\frac{6 x}{x - 4}) = \frac{2 {(x + 8)}^{2}}{9 x (x - 4)}$

Find f(g(x)) f(x)=(x^2-8x+16)/(x+2x) , g(x)=(6x)/(x-4)

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