Algebra Examples

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

Step 1

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

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

Step 2

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

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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.

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

Step 2.1.2

Combine the numerators over the common denominator.

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2.1.2

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

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

Step 2.2.2

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

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

Step 2.2.3

Combine the numerators over the common denominator.

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

Step 2.2.4

Simplify the numerator.

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Step 2.2.4.1

Factor $x$ out of $x (x - 1) + x$ .

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Step 2.2.4.1.1

Raise $x$ to the power of $1$ .

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

Step 2.2.4.1.2

Factor $x$ out of $x^{1}$ .

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

Step 2.2.4.1.3

Factor $x$ out of $x (x - 1) + x \cdot 1$ .

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

Step 2.2.4.2

Add $- 1$ and $1$ .

$y = \frac{1}{\frac{x (x + 0)}{x - 1}}$

Step 2.2.4.3

Add $x$ and $0$ .

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

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Raise $x$ to the power of $1$ .

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

Step 2.2.5.2

Raise $x$ to the power of $1$ .

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

Step 2.2.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.5.4

Add $1$ and $1$ .

$y = \frac{1}{\frac{x^{2}}{x - 1}}$

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

$y = 1 \frac{x - 1}{x^{2}}$

Step 2.4

Multiply $\frac{x - 1}{x^{2}}$ by $1$ .

$y = \frac{x - 1}{x^{2}}$