Find the Inverse (2x-1)/x

$\frac{2 x - 1}{x}$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{2 y - 1}{y} = x$ .

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

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 $\frac{2 y - 1}{y} = x$ by $y$ to eliminate the fractions.

Multiply each term in $\frac{2 y - 1}{y} = x$ by $y$ .

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

Simplify the left side.

Cancel the common factor of $y$ .

Cancel the common factor.

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

Rewrite the expression.

$2 y - 1 = x y$

Solve the equation.

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

$2 y - 1 - x y = 0$

Add $1$ to both sides of the equation.

$2 y - x y = 1$

Factor $y$ out of $2 y - x y$ .

Factor $y$ out of $2 y$ .

$y \cdot 2 - x y = 1$

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

$y \cdot 2 + y (- x) = 1$

Factor $y$ out of $y \cdot 2 + y (- x)$ .

$y (2 - x) = 1$

Divide each term in $y (2 - x) = 1$ by $2 - x$ and simplify.

Divide each term in $y (2 - x) = 1$ by $2 - x$ .

$\frac{y (2 - x)}{2 - x} = \frac{1}{2 - x}$

Simplify the left side.

Cancel the common factor of $2 - x$ .

Cancel the common factor.

$\frac{y (2 - x)}{2 - x} = \frac{1}{2 - x}$

Divide $y$ by $1$ .

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

Step 3

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

$f^{- 1} (x) = \frac{1}{2 - x}$

Step 4

Verify if $f^{- 1} (x) = \frac{1}{2 - x}$ is the inverse of $f (x) = \frac{2 x - 1}{x}$ .

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 $f^{- 1} (\frac{2 x - 1}{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2 x - 1}{x}) = \frac{1}{2 - (\frac{2 x - 1}{x})}$

Simplify the denominator.

To write $2$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$f^{- 1} (\frac{2 x - 1}{x}) = \frac{1}{\frac{2 x}{x} - \frac{2 x - 1}{x}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{2 x - 1}{x}) = \frac{1}{\frac{2 x - (2 x - 1)}{x}}$

Rewrite $\frac{2 x - (2 x - 1)}{x}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{2 x - 1}{x}) = \frac{1}{\frac{2 x - (2 x) + 1}{x}}$

Multiply $2$ by $- 1$ .

$f^{- 1} (\frac{2 x - 1}{x}) = \frac{1}{\frac{2 x - 2 x + 1}{x}}$

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{2 x - 1}{x}) = \frac{1}{\frac{2 x - 2 x + 1}{x}}$

Subtract $2 x$ from $2 x$ .

$f^{- 1} (\frac{2 x - 1}{x}) = \frac{1}{\frac{0 + 1}{x}}$

Add $0$ and $1$ .

$f^{- 1} (\frac{2 x - 1}{x}) = \frac{1}{\frac{1}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{2 x - 1}{x}) = 1 x$

Multiply $x$ by $1$ .

$f^{- 1} (\frac{2 x - 1}{x}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{1}{2 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{1}{2 - x}) = \frac{2 (\frac{1}{2 - x}) - 1}{\frac{1}{2 - x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{1}{2 - x}) = (2 (\frac{1}{2 - x}) - 1) (2 - x)$

Combine $2$ and $\frac{1}{2 - x}$ .

$f (\frac{1}{2 - x}) = (\frac{2}{2 - x} - 1) (2 - x)$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2 - x}{2 - x}$ .

$f (\frac{1}{2 - x}) = (\frac{2}{2 - x} - 1 \cdot \frac{2 - x}{2 - x}) (2 - x)$

Simplify terms.

Combine $- 1$ and $\frac{2 - x}{2 - x}$ .

$f (\frac{1}{2 - x}) = (\frac{2}{2 - x} + \frac{- (2 - x)}{2 - x}) (2 - x)$

Combine the numerators over the common denominator.

$f (\frac{1}{2 - x}) = \frac{2 - (2 - x)}{2 - x} \cdot (2 - x)$

Simplify the numerator.

Apply the distributive property.

$f (\frac{1}{2 - x}) = \frac{2 - 1 \cdot 2 + x}{2 - x} \cdot (2 - x)$

Multiply $- 1$ by $2$ .

$f (\frac{1}{2 - x}) = \frac{2 - 2 + x}{2 - x} \cdot (2 - x)$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$f (\frac{1}{2 - x}) = \frac{2 - 2 + 1 x}{2 - x} \cdot (2 - x)$

Multiply $x$ by $1$ .

$f (\frac{1}{2 - x}) = \frac{2 - 2 + x}{2 - x} \cdot (2 - x)$

Subtract $2$ from $2$ .

$f (\frac{1}{2 - x}) = \frac{0 + x}{2 - x} \cdot (2 - x)$

Add $0$ and $x$ .

$f (\frac{1}{2 - x}) = \frac{x}{2 - x} \cdot (2 - x)$

Cancel the common factor of $2 - x$ .

Cancel the common factor.

$f (\frac{1}{2 - x}) = \frac{x}{2 - x} \cdot (2 - x)$

Rewrite the expression.

$f (\frac{1}{2 - x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{1}{2 - x}$ is the inverse of $f (x) = \frac{2 x - 1}{x}$ .

$f^{- 1} (x) = \frac{1}{2 - x}$

Find the Inverse (2x-1)/x

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