Algebra Examples

$f (x) = \frac{11 + x}{11 - x}$ , $g (x) = \frac{11 x}{11 - x}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (\frac{11 + x}{11 - x})$ by substituting in the value of $f$ into $g$ .

$g (\frac{11 + x}{11 - x}) = \frac{11 (\frac{11 + x}{11 - x})}{11 - (\frac{11 + x}{11 - x})}$

Step 3

Combine $11$ and $\frac{11 + x}{11 - x}$ .

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{11 - \frac{11 + x}{11 - x}}$

Step 4

Simplify the denominator.

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Step 4.1

To write $11$ as a fraction with a common denominator, multiply by $\frac{11 - x}{11 - x}$ .

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{11 (11 - x)}{11 - x} - \frac{11 + x}{11 - x}}$

Step 4.2

Combine the numerators over the common denominator.

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{11 (11 - x) - (11 + x)}{11 - x}}$

Step 4.3

Reorder terms.

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{11 (11 - x) - (11 + x)}{- x + 11}}$

Step 4.4

Rewrite $\frac{11 (11 - x) - (11 + x)}{- x + 11}$ in a factored form.

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Step 4.4.1

Apply the distributive property.

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{11 \cdot 11 + 11 (- x) - (11 + x)}{- x + 11}}$

Step 4.4.2

Multiply $11$ by $11$ .

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{121 + 11 (- x) - (11 + x)}{- x + 11}}$

Step 4.4.3

Multiply $- 1$ by $11$ .

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{121 - 11 x - (11 + x)}{- x + 11}}$

Step 4.4.4

Apply the distributive property.

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{121 - 11 x - 1 \cdot 11 - x}{- x + 11}}$

Step 4.4.5

Multiply $- 1$ by $11$ .

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{121 - 11 x - 11 - x}{- x + 11}}$

Step 4.4.6

Subtract $11$ from $121$ .

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{- 11 x + 110 - x}{- x + 11}}$

Step 4.4.7

Subtract $x$ from $- 11 x$ .

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{- 12 x + 110}{- x + 11}}$

Step 4.4.8

Factor $2$ out of $- 12 x + 110$ .

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Step 4.4.8.1

Factor $2$ out of $- 12 x$ .

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{2 (- 6 x) + 110}{- x + 11}}$

Step 4.4.8.2

Factor $2$ out of $110$ .

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{2 (- 6 x) + 2 (55)}{- x + 11}}$

Step 4.4.8.3

Factor $2$ out of $2 (- 6 x) + 2 (55)$ .

$g (\frac{11 + x}{11 - x}) = \frac{\frac{11 (11 + x)}{11 - x}}{\frac{2 (- 6 x + 55)}{- x + 11}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{11 + x}{11 - x}) = \frac{11 (11 + x)}{11 - x} \cdot \frac{- x + 11}{2 (- 6 x + 55)}$

Step 6

Multiply $\frac{11 (11 + x)}{11 - x}$ by $\frac{- x + 11}{2 (- 6 x + 55)}$ .

$g (\frac{11 + x}{11 - x}) = \frac{11 (11 + x) (- x + 11)}{(11 - x) (2 (- 6 x + 55))}$

Step 7

Cancel the common factor of $- x + 11$ and $11 - x$ .

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Step 7.1

Reorder terms.

$g (\frac{11 + x}{11 - x}) = \frac{11 (11 + x) (- x + 11)}{(- x + 11) (2 (- 6 x + 55))}$

Step 7.2

Cancel the common factor.

$g (\frac{11 + x}{11 - x}) = \frac{11 (11 + x) (- x + 11)}{(- x + 11) (2 (- 6 x + 55))}$

Step 7.3

Rewrite the expression.

$g (\frac{11 + x}{11 - x}) = \frac{11 (11 + x)}{2 (- 6 x + 55)}$

Step 8

Factor $- 1$ out of $- 6 x$ .

$g (\frac{11 + x}{11 - x}) = \frac{11 (11 + x)}{2 (- (6 x) + 55)}$

Step 9

Rewrite $55$ as $- 1 (- 55)$ .

$g (\frac{11 + x}{11 - x}) = \frac{11 (11 + x)}{2 (- (6 x) - 1 \cdot - 55)}$

Step 10

Factor $- 1$ out of $- (6 x) - 1 (- 55)$ .

$g (\frac{11 + x}{11 - x}) = \frac{11 (11 + x)}{2 (- (6 x - 55))}$

Step 11

Rewrite negatives.

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Step 11.1

Rewrite $- (6 x - 55)$ as $- 1 (6 x - 55)$ .

$g (\frac{11 + x}{11 - x}) = \frac{11 (11 + x)}{2 (- 1 (6 x - 55))}$

Step 11.2

Move the negative in front of the fraction.

$g (\frac{11 + x}{11 - x}) = - \frac{11 (11 + x)}{2 (6 x - 55)}$