Algebra Examples

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

Step 1

Write $f (x) = \frac{1}{1 - \frac{1}{1 + \frac{1}{x}}}$ as an equation.

$y = \frac{1}{1 - \frac{1}{1 + \frac{1}{x}}}$

Step 2

Simplify $\frac{1}{1 - \frac{1}{1 + \frac{1}{x}}}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Write $1$ as a fraction with a common denominator.

$y = \frac{1}{1 - \frac{1}{\frac{x}{x} + \frac{1}{x}}}$

Step 2.1.2

Combine the numerators over the common denominator.

$y = \frac{1}{1 - \frac{1}{\frac{x + 1}{x}}}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2.2

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

$y = \frac{1}{1 - \frac{x}{x + 1}}$

Step 2.2.3

Write $1$ as a fraction with a common denominator.

$y = \frac{1}{\frac{x + 1}{x + 1} - \frac{x}{x + 1}}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \frac{1}{\frac{x + 1 - x}{x + 1}}$

Step 2.2.5

Reorder terms.

$y = \frac{1}{\frac{x - x + 1}{x + 1}}$

Step 2.2.6

Rewrite $\frac{x - x + 1}{x + 1}$ in a factored form.

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Step 2.2.6.1

Subtract $x$ from $x$ .

$y = \frac{1}{\frac{0 + 1}{x + 1}}$

Step 2.2.6.2

Add $0$ and $1$ .

$y = \frac{1}{\frac{1}{x + 1}}$

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

$y = 1 (x + 1)$

Step 2.4

Multiply $x + 1$ by $1$ .

$y = x + 1$