Algebra Examples

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Remove parentheses.

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

Step 3.2

Cancel the common factor of $11 + \frac{11 x}{11 - x}$ and $11 - (\frac{11 x}{11 - x})$ .

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Step 3.2.1

Factor $11$ out of $11$ .

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

Step 3.2.2

Factor $11$ out of $\frac{11 x}{11 - x}$ .

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

Step 3.2.3

Factor $11$ out of $11 (1) + 11 \frac{x}{11 - x}$ .

$f (\frac{11 x}{11 - x}) = \frac{11 (1 + \frac{x}{11 - x})}{11 - \frac{11 x}{11 - x}}$

Step 3.2.4

Cancel the common factors.

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Step 3.2.4.1

Factor $11$ out of $11$ .

$f (\frac{11 x}{11 - x}) = \frac{11 (1 + \frac{x}{11 - x})}{11 (1) - \frac{11 x}{11 - x}}$

Step 3.2.4.2

Factor $11$ out of $- \frac{11 x}{11 - x}$ .

$f (\frac{11 x}{11 - x}) = \frac{11 (1 + \frac{x}{11 - x})}{11 (1) + 11 (- \frac{x}{11 - x})}$

Step 3.2.4.3

Factor $11$ out of $11 (1) + 11 (- \frac{x}{11 - x})$ .

$f (\frac{11 x}{11 - x}) = \frac{11 (1 + \frac{x}{11 - x})}{11 (1 - \frac{x}{11 - x})}$

Step 3.2.4.4

Cancel the common factor.

$f (\frac{11 x}{11 - x}) = \frac{11 (1 + \frac{x}{11 - x})}{11 (1 - \frac{x}{11 - x})}$

Step 3.2.4.5

Rewrite the expression.

$f (\frac{11 x}{11 - x}) = \frac{1 + \frac{x}{11 - x}}{1 - \frac{x}{11 - x}}$

Step 4

Multiply the numerator and denominator of the fraction by $11 - x$ .

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Step 4.1

Multiply $\frac{1 + \frac{x}{11 - x}}{1 - \frac{x}{11 - x}}$ by $\frac{11 - x}{11 - x}$ .

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

Step 4.2

Combine.

$f (\frac{11 x}{11 - x}) = \frac{(11 - x) (1 + \frac{x}{11 - x})}{(11 - x) (1 - \frac{x}{11 - x})}$

Step 5

Apply the distributive property.

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

Step 6

Simplify by cancelling.

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Step 6.1

Cancel the common factor of $11 - x$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $11 - x$ .

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Step 6.2.1

Move the leading negative in $- \frac{x}{11 - x}$ into the numerator.

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $11 - x$ by $1$ .

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

Step 7.2

Add $- x$ and $x$ .

$f (\frac{11 x}{11 - x}) = \frac{11 + 0}{(11 - x) \cdot 1 - x}$

Step 7.3

Add $11$ and $0$ .

$f (\frac{11 x}{11 - x}) = \frac{11}{(11 - x) \cdot 1 - x}$

Step 8

Simplify the denominator.

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Step 8.1

Multiply $11 - x$ by $1$ .

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

Step 8.2

Subtract $x$ from $- x$ .

$f (\frac{11 x}{11 - x}) = \frac{11}{11 - 2 x}$