Find the Inverse (x+20)/x

$\frac{x + 20}{x}$

Step 1

Interchange the variables.

$x = \frac{y + 20}{y}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y + 20}{y} = x$ .

$\frac{y + 20}{y} = x$

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 $\frac{y + 20}{y} = x$ by $y$ to eliminate the fractions.

Multiply each term in $\frac{y + 20}{y} = x$ by $y$ .

$\frac{y + 20}{y} y = x y$

Simplify the left side.

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{y + 20}{y} y = x y$

Rewrite the expression.

$y + 20 = x y$

Solve the equation.

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

$y + 20 - x y = 0$

Subtract $20$ from both sides of the equation.

$y - x y = - 20$

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

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

$y \cdot 1 - x y = - 20$

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

$y \cdot 1 + y (- x) = - 20$

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

$y (1 - x) = - 20$

Divide each term in $y (1 - x) = - 20$ by $1 - x$ and simplify.

Divide each term in $y (1 - x) = - 20$ by $1 - x$ .

$\frac{y (1 - x)}{1 - x} = \frac{- 20}{1 - x}$

Simplify the left side.

Cancel the common factor of $1 - x$ .

Cancel the common factor.

$\frac{y (1 - x)}{1 - x} = \frac{- 20}{1 - x}$

Divide $y$ by $1$ .

$y = \frac{- 20}{1 - x}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{20}{1 - x}$

Step 3

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

$f^{- 1} (x) = - \frac{20}{1 - x}$

Step 4

Verify if $f^{- 1} (x) = - \frac{20}{1 - x}$ is the inverse of $f (x) = \frac{x + 20}{x}$ .

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 $f^{- 1} (\frac{x + 20}{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x + 20}{x}) = - \frac{20}{1 - (\frac{x + 20}{x})}$

Simplify the denominator.

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

$f^{- 1} (\frac{x + 20}{x}) = - \frac{20}{\frac{x}{x} - \frac{x + 20}{x}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 20}{x}) = - \frac{20}{\frac{x - (x + 20)}{x}}$

Rewrite $\frac{x - (x + 20)}{x}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{x + 20}{x}) = - \frac{20}{\frac{x - x - 1 \cdot 20}{x}}$

Multiply $- 1$ by $20$ .

$f^{- 1} (\frac{x + 20}{x}) = - \frac{20}{\frac{x - x - 20}{x}}$

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x + 20}{x}) = - \frac{20}{\frac{0 - 20}{x}}$

Subtract $20$ from $0$ .

$f^{- 1} (\frac{x + 20}{x}) = - \frac{20}{\frac{- 20}{x}}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{x + 20}{x}) = - \frac{20}{- \frac{20}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{x + 20}{x}) = - (20 (- \frac{x}{20}))$

Cancel the common factor of $20$ .

Move the leading negative in $- \frac{x}{20}$ into the numerator.

$f^{- 1} (\frac{x + 20}{x}) = - (20 (\frac{- x}{20}))$

Cancel the common factor.

$f^{- 1} (\frac{x + 20}{x}) = - (20 (\frac{- x}{20}))$

Rewrite the expression.

$f^{- 1} (\frac{x + 20}{x}) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{20}{1 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{20}{1 - x}) = \frac{(- \frac{20}{1 - x}) + 20}{- \frac{20}{1 - x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{20}{1 - x}) = (- \frac{20}{1 - x} + 20) (- \frac{1 - x}{20})$

To write $20$ as a fraction with a common denominator, multiply by $\frac{1 - x}{1 - x}$ .

$f (- \frac{20}{1 - x}) = (- \frac{20}{1 - x} + \frac{20 (1 - x)}{1 - x}) (- \frac{1 - x}{20})$

Simplify terms.

Combine the numerators over the common denominator.

$f (- \frac{20}{1 - x}) = \frac{- 20 + 20 (1 - x)}{1 - x} \cdot (- \frac{1 - x}{20})$

Rewrite using the commutative property of multiplication.

$f (- \frac{20}{1 - x}) = - \frac{- 20 + 20 (1 - x)}{1 - x} \cdot \frac{1 - x}{20}$

Cancel the common factor of $1 - x$ .

Move the leading negative in $- \frac{- 20 + 20 (1 - x)}{1 - x}$ into the numerator.

$f (- \frac{20}{1 - x}) = \frac{- (- 20 + 20 (1 - x))}{1 - x} \cdot \frac{1 - x}{20}$

Cancel the common factor.

$f (- \frac{20}{1 - x}) = \frac{- (- 20 + 20 (1 - x))}{1 - x} \cdot \frac{1 - x}{20}$

Rewrite the expression.

$f (- \frac{20}{1 - x}) = - (- 20 + 20 (1 - x)) \frac{1}{20}$

Simplify each term.

Apply the distributive property.

$f (- \frac{20}{1 - x}) = - (- 20 + 20 \cdot 1 + 20 (- x)) \frac{1}{20}$

Multiply $20$ by $1$ .

$f (- \frac{20}{1 - x}) = - (- 20 + 20 + 20 (- x)) \frac{1}{20}$

Multiply $- 1$ by $20$ .

$f (- \frac{20}{1 - x}) = - (- 20 + 20 - 20 x) \frac{1}{20}$

Reduce the expression by cancelling the common factors.

Add $- 20$ and $20$ .

$f (- \frac{20}{1 - x}) = - (0 - 20 x) \frac{1}{20}$

Subtract $20 x$ from $0$ .

$f (- \frac{20}{1 - x}) = 20 x \frac{1}{20}$

Cancel the common factor of $20$ .

Factor $20$ out of $- (- 20 x)$ .

$f (- \frac{20}{1 - x}) = 20 (x) (\frac{1}{20})$

Cancel the common factor.

$f (- \frac{20}{1 - x}) = 20 (x) (\frac{1}{20})$

Rewrite the expression.

$f (- \frac{20}{1 - x}) = x$

Multiply.

Multiply $- 1$ by $- 1$ .

$f (- \frac{20}{1 - x}) = 1 x$

Multiply $x$ by $1$ .

$f (- \frac{20}{1 - x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{20}{1 - x}$ is the inverse of $f (x) = \frac{x + 20}{x}$ .

$f^{- 1} (x) = - \frac{20}{1 - x}$

Find the Inverse (x+20)/x

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