Algebra Examples

$f (x) = \frac{{(x - 4)}^{2}}{(x + 2 x)}$ ; $g (x) = \frac{(2 x)}{(x - 4)}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (\frac{2 x}{x - 4})$ by substituting in the value of $g$ into $f$ .

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Rewrite $\frac{2 x - 4 (x - 4)}{x - 4}$ in a factored form.

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Step 3.4.1

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

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Step 3.4.1.1

Factor $2$ out of $2 x$ .

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

Step 3.4.1.2

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

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

Step 3.4.1.3

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

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

Step 3.4.2

Apply the distributive property.

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

Step 3.4.3

Multiply $- 2$ by $- 4$ .

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

Step 3.4.4

Subtract $2 x$ from $x$ .

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

Step 3.5

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

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 4

Simplify the denominator.

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Step 4.1

Multiply $2 \frac{2 x}{x - 4}$ .

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Step 4.1.1

Combine $2$ and $\frac{2 x}{x - 4}$ .

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

Step 4.1.2

Multiply $2$ by $2$ .

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 4.3

Reorder terms.

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

Step 4.4

Add $4 x$ and $2 x$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $2$ .

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Step 6.1

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

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

Step 6.2

Factor $2$ out of $6 x$ .

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

Step 6.3

Cancel the common factor.

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

Step 6.4

Rewrite the expression.

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

Step 7

Cancel the common factor of $x - 4$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

$f (\frac{2 x}{x - 4}) = \frac{2 {(- x + 8)}^{2}}{x - 4} \cdot \frac{1}{3 x}$

Step 8

Multiply $\frac{2 {(- x + 8)}^{2}}{x - 4}$ by $\frac{1}{3 x}$ .

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

Step 9

Simplify the expression.

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Step 9.1

Move $3$ to the left of $x - 4$ .

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

Step 9.2

Reorder factors in $\frac{2 {(- x + 8)}^{2}}{3 (x - 4) x}$ .

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