Find the Inverse y=(x+1)/x

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

Interchange the variables.

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

Solve for $y$ .

Rewrite the equation as $\frac{y + 1}{y} = x$ .

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

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1$

The LCM of one and any expression is the expression.

$y$

Multiply each term in $\frac{y + 1}{y} = x$ by $y$ to eliminate the fractions.

Multiply each term in $\frac{y + 1}{y} = x$ by $y$ .

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

Simplify the left side.

Cancel the common factor of $y$ .

Cancel the common factor.

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

Rewrite the expression.

$y + 1 = x y$

Solve the equation.

Subtract $x y$ from both sides of the equation.

$y + 1 - x y = 0$

Subtract $1$ from both sides of the equation.

$y - x y = - 1$

Factor $y$ out of $y - x y$ .

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

$y \cdot 1 - x y = - 1$

Factor $y$ out of $- x y$ .

$y \cdot 1 + y (- x) = - 1$

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

$y (1 - x) = - 1$

Divide each term in $y (1 - x) = - 1$ by $1 - x$ and simplify.

Divide each term in $y (1 - x) = - 1$ by $1 - x$ .

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

Simplify the left side.

Cancel the common factor of $1 - x$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

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

Verify if $f^{- 1} (x) = - \frac{1}{1 - x}$ is the inverse of $f (x) = \frac{x + 1}{x}$ .

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (\frac{x + 1}{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Simplify the denominator.

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

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

Combine the numerators over the common denominator.

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

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $x$ from $x$ .

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

Subtract $1$ from $0$ .

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

Move the negative in front of the fraction.

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

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $x$ by $1$ .

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

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (- \frac{1}{1 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

Combine the numerators over the common denominator.

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

Rewrite using the commutative property of multiplication.

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

Cancel the common factor of $1 - x$ .

Move the leading negative in $- \frac{- 1 + 1 - x}{1 - x}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify by adding numbers.

Add $- 1$ and $1$ .

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

Subtract $x$ from $0$ .

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

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{1}{1 - x}$ is the inverse of $f (x) = \frac{x + 1}{x}$ .

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

Find the Inverse y=(x+1)/x

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