Algebra Examples

$f (x) = \frac{x - 1}{x}$ $g (x) = \frac{x}{x + 2}$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

$f (\frac{x}{x + 2}) = \frac{(\frac{x}{x + 2}) - 1}{\frac{x}{x + 2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{x}{x + 2}) = (\frac{x}{x + 2} - 1) (\frac{x + 2}{x})$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$f (\frac{x}{x + 2}) = (\frac{x}{x + 2} - 1 \cdot \frac{x + 2}{x + 2}) (\frac{x + 2}{x})$

Step 5

Simplify terms.

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Step 5.1

Combine $- 1$ and $\frac{x + 2}{x + 2}$ .

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$f (\frac{x}{x + 2}) = \frac{x - x - 1 \cdot 2}{x + 2} \cdot \frac{x + 2}{x}$

Step 6.2

Multiply $- 1$ by $2$ .

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

Step 6.3

Subtract $x$ from $x$ .

$f (\frac{x}{x + 2}) = \frac{0 - 2}{x + 2} \cdot \frac{x + 2}{x}$

Step 6.4

Subtract $2$ from $0$ .

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

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $x + 2$ .

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Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Rewrite the expression.

$f (\frac{x}{x + 2}) = - 2 \frac{1}{x}$

Step 7.2

Combine $- 2$ and $\frac{1}{x}$ .

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

Step 7.3

Move the negative in front of the fraction.

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