Simplify f(x)=1/(1-1/(1+1/x))

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

Step 1

Simplify the denominator.

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

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

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Reorder terms.

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

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

Subtract $x$ from $x$ .

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

Add $0$ and $1$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

$f (x) = 1 (x + 1)$

Step 4

Multiply $x + 1$ by $1$ .

$f (x) = x + 1$