Find the Inverse y=(x-2)/(3x-1)

$y = \frac{x - 2}{3 x - 1}$

Interchange the variables.

$x = \frac{y - 2}{3 y - 1}$

Solve for $y$ .

Rewrite the equation as $\frac{y - 2}{3 y - 1} = x$ .

$\frac{y - 2}{3 y - 1} = x$

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.

$3 y - 1, 1$

Remove parentheses.

$3 y - 1, 1$

The LCM of one and any expression is the expression.

$3 y - 1$

Multiply each term in $\frac{y - 2}{3 y - 1} = x$ by $3 y - 1$ to eliminate the fractions.

Multiply each term in $\frac{y - 2}{3 y - 1} = x$ by $3 y - 1$ .

$\frac{y - 2}{3 y - 1} (3 y - 1) = x (3 y - 1)$

Simplify the left side.

Cancel the common factor of $3 y - 1$ .

Cancel the common factor.

$\frac{y - 2}{3 y - 1} (3 y - 1) = x (3 y - 1)$

Rewrite the expression.

$y - 2 = x (3 y - 1)$

Simplify the right side.

Simplify by multiplying through.

Apply the distributive property.

$y - 2 = x (3 y) + x \cdot - 1$

Reorder.

Rewrite using the commutative property of multiplication.

$y - 2 = 3 x y + x \cdot - 1$

Move $- 1$ to the left of $x$ .

$y - 2 = 3 x y - 1 \cdot x$

Rewrite $- 1 x$ as $- x$ .

$y - 2 = 3 x y - x$

Solve the equation.

Subtract $3 x y$ from both sides of the equation.

$y - 2 - 3 x y = - x$

Add $2$ to both sides of the equation.

$y - 3 x y = - x + 2$

Factor $y$ out of $y - 3 x y$ .

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

$y \cdot 1 - 3 x y = - x + 2$

Factor $y$ out of $- 3 x y$ .

$y \cdot 1 + y (- 3 x) = - x + 2$

Factor $y$ out of $y \cdot 1 + y (- 3 x)$ .

$y (1 - 3 x) = - x + 2$

Divide each term in $y (1 - 3 x) = - x + 2$ by $1 - 3 x$ and simplify.

Divide each term in $y (1 - 3 x) = - x + 2$ by $1 - 3 x$ .

$\frac{y (1 - 3 x)}{1 - 3 x} = \frac{- x}{1 - 3 x} + \frac{2}{1 - 3 x}$

Simplify the left side.

Cancel the common factor of $1 - 3 x$ .

Cancel the common factor.

$\frac{y (1 - 3 x)}{1 - 3 x} = \frac{- x}{1 - 3 x} + \frac{2}{1 - 3 x}$

Divide $y$ by $1$ .

$y = \frac{- x}{1 - 3 x} + \frac{2}{1 - 3 x}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{- x + 2}{1 - 3 x}$

Factor $- 1$ out of $- x$ .

$y = \frac{- (x) + 2}{1 - 3 x}$

Rewrite $2$ as $- 1 (- 2)$ .

$y = \frac{- (x) - 1 (- 2)}{1 - 3 x}$

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$y = \frac{- (x - 2)}{1 - 3 x}$

Simplify the expression.

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$y = \frac{- 1 (x - 2)}{1 - 3 x}$

Move the negative in front of the fraction.

$y = - \frac{x - 2}{1 - 3 x}$

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

$f^{- 1} (x) = - \frac{x - 2}{1 - 3 x}$

Verify if $f^{- 1} (x) = - \frac{x - 2}{1 - 3 x}$ is the inverse of $f (x) = \frac{x - 2}{3 x - 1}$ .

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 $f^{- 1} (\frac{x - 2}{3 x - 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{(\frac{x - 2}{3 x - 1}) - 2}{1 - 3 \frac{x - 2}{3 x - 1}}$

Multiply the numerator and denominator of the complex fraction by $3 x - 1$ .

Multiply $\frac{\frac{x - 2}{3 x - 1} - 2}{1 - 3 \frac{x - 2}{3 x - 1}}$ by $\frac{3 x - 1}{3 x - 1}$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - (\frac{3 x - 1}{3 x - 1} \cdot \frac{\frac{x - 2}{3 x - 1} - 2}{1 - 3 \frac{x - 2}{3 x - 1}})$

Combine.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{(3 x - 1) (\frac{x - 2}{3 x - 1} - 2)}{(3 x - 1) (1 - 3 \frac{x - 2}{3 x - 1})}$

Apply the distributive property.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{(3 x - 1) (\frac{x - 2}{3 x - 1}) + (3 x - 1) \cdot - 2}{(3 x - 1) \cdot 1 + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})}$

Cancel the common factor of $3 x - 1$ .

Cancel the common factor.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{(3 x - 1) (\frac{x - 2}{3 x - 1}) + (3 x - 1) \cdot - 2}{(3 x - 1) \cdot 1 + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})}$

Rewrite the expression.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{x - 2 + (3 x - 1) \cdot - 2}{(3 x - 1) \cdot 1 + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})}$

Simplify the numerator.

Apply the distributive property.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{x - 2 + 3 x \cdot - 2 - 1 \cdot - 2}{(3 x - 1) \cdot 1 + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})}$

Multiply $- 2$ by $3$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{x - 2 - 6 x - 1 \cdot - 2}{(3 x - 1) \cdot 1 + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})}$

Multiply $- 1$ by $- 2$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{x - 2 - 6 x + 2}{(3 x - 1) \cdot 1 + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})}$

Subtract $6 x$ from $x$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x - 2 + 2}{(3 x - 1) \cdot 1 + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})}$

Add $- 2$ and $2$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x + 0}{(3 x - 1) \cdot 1 + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})}$

Add $- 5 x$ and $0$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) \cdot 1 + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})}$

Simplify the denominator.

Factor $3 x - 1$ out of $(3 x - 1) \cdot 1 + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})$ .

Factor $3 x - 1$ out of $(3 x - 1) \cdot 1$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (1) + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})}$

Factor $3 x - 1$ out of $(3 x - 1) (1) + (3 x - 1) (- 3 \frac{x - 2}{3 x - 1})$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (1 - 3 \frac{x - 2}{3 x - 1})}$

Combine $- 3$ and $\frac{x - 2}{3 x - 1}$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (1 + \frac{- 3 (x - 2)}{3 x - 1})}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (1 - \frac{(3) (x - 2)}{3 x - 1})}$

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

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (\frac{3 x - 1}{3 x - 1} - \frac{3 (x - 2)}{3 x - 1})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (\frac{3 x - 1 - 3 (x - 2)}{3 x - 1})}$

Rewrite $\frac{3 x - 1 - 3 (x - 2)}{3 x - 1}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (\frac{3 x - 1 - 3 x - 3 \cdot - 2}{3 x - 1})}$

Multiply $- 3$ by $- 2$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (\frac{3 x - 1 - 3 x + 6}{3 x - 1})}$

Subtract $3 x$ from $3 x$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (\frac{0 - 1 + 6}{3 x - 1})}$

Subtract $1$ from $0$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (\frac{- 1 + 6}{3 x - 1})}$

Add $- 1$ and $6$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - \frac{- 5 x}{(3 x - 1) (\frac{5}{3 x - 1})}$

Factor $\frac{5}{3 x - 1}$ out of $\frac{- 5 x}{(3 x - 1) \frac{5}{3 x - 1}}$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - (\frac{3 x - 1}{5} \cdot \frac{- 5 x}{3 x - 1})$

Cancel the common factor of $3 x - 1$ .

Cancel the common factor.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - (\frac{3 x - 1}{5} \cdot \frac{- 5 x}{3 x - 1})$

Rewrite the expression.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - (\frac{1}{5} \cdot (- 5 x))$

Cancel the common factor of $5$ .

Factor $5$ out of $- 5 x$ .

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - (\frac{1}{5} \cdot (5 (- x)))$

Cancel the common factor.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = - (\frac{1}{5} \cdot (5 (- x)))$

Rewrite the expression.

$f^{- 1} (\frac{x - 2}{3 x - 1}) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{x - 2}{1 - 3 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{(- \frac{x - 2}{1 - 3 x}) - 2}{3 (- \frac{x - 2}{1 - 3 x}) - 1}$

Multiply the numerator and denominator of the complex fraction by $1 - 3 x$ .

Multiply $\frac{- \frac{x - 2}{1 - 3 x} - 2}{3 (- \frac{x - 2}{1 - 3 x}) - 1}$ by $\frac{1 - 3 x}{1 - 3 x}$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{1 - 3 x}{1 - 3 x} \cdot \frac{- \frac{x - 2}{1 - 3 x} - 2}{3 (- \frac{x - 2}{1 - 3 x}) - 1}$

Combine.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{(1 - 3 x) (- \frac{x - 2}{1 - 3 x} - 2)}{(1 - 3 x) (3 (- \frac{x - 2}{1 - 3 x}) - 1)}$

Apply the distributive property.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{(1 - 3 x) (- \frac{x - 2}{1 - 3 x}) + (1 - 3 x) \cdot - 2}{(1 - 3 x) (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Cancel the common factor of $1 - 3 x$ .

Move the leading negative in $- \frac{x - 2}{1 - 3 x}$ into the numerator.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{(1 - 3 x) (\frac{- (x - 2)}{1 - 3 x}) + (1 - 3 x) \cdot - 2}{(1 - 3 x) (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Cancel the common factor.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{(1 - 3 x) (\frac{- (x - 2)}{1 - 3 x}) + (1 - 3 x) \cdot - 2}{(1 - 3 x) (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Rewrite the expression.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{- (x - 2) + (1 - 3 x) \cdot - 2}{(1 - 3 x) (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Simplify the numerator.

Apply the distributive property.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{- x + 2 + (1 - 3 x) \cdot - 2}{(1 - 3 x) \cdot (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Multiply $- 1$ by $- 2$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{- x + 2 + (1 - 3 x) \cdot - 2}{(1 - 3 x) \cdot (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Apply the distributive property.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{- x + 2 + 1 \cdot - 2 - 3 x \cdot - 2}{(1 - 3 x) \cdot (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Multiply $- 2$ by $1$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{- x + 2 - 2 - 3 x \cdot - 2}{(1 - 3 x) \cdot (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Multiply $- 2$ by $- 3$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{- x + 2 - 2 + 6 x}{(1 - 3 x) \cdot (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Add $- x$ and $6 x$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x + 2 - 2}{(1 - 3 x) \cdot (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Subtract $2$ from $2$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x + 0}{(1 - 3 x) \cdot (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Add $5 x$ and $0$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) \cdot (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) \cdot - 1}$

Simplify the denominator.

Factor $1 - 3 x$ out of $(1 - 3 x) \cdot 3 (- \frac{x - 2}{1 - 3 x}) + (1 - 3 x) \cdot - 1$ .

Factor $1 - 3 x$ out of $(1 - 3 x) \cdot - 1$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) (- 1)}$

Factor $1 - 3 x$ out of $(1 - 3 x) (3 (- \frac{x - 2}{1 - 3 x})) + (1 - 3 x) (- 1)$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (3 (- \frac{x - 2}{1 - 3 x}) - 1)}$

Multiply $3 (- \frac{x - 2}{1 - 3 x})$ .

Multiply $- 1$ by $3$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (- 3 \frac{x - 2}{1 - 3 x} - 1)}$

Combine $- 3$ and $\frac{x - 2}{1 - 3 x}$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{- 3 (x - 2)}{1 - 3 x} - 1)}$

Move the negative in front of the fraction.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (- \frac{(3) (x - 2)}{1 - 3 x} - 1)}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{1 - 3 x}{1 - 3 x}$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (- \frac{3 (x - 2)}{1 - 3 x} - 1 \cdot \frac{1 - 3 x}{1 - 3 x})}$

Combine $- 1$ and $\frac{1 - 3 x}{1 - 3 x}$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (- \frac{3 (x - 2)}{1 - 3 x} + \frac{- (1 - 3 x)}{1 - 3 x})}$

Combine the numerators over the common denominator.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{- 3 (x - 2) - (1 - 3 x)}{1 - 3 x})}$

Rewrite $\frac{- 3 (x - 2) - (1 - 3 x)}{1 - 3 x}$ in a factored form.

Factor $- 1$ out of $- 3 (x - 2) - (1 - 3 x)$ .

Reorder $1$ and $- 3 x$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{- 3 (x - 2) - (- 3 x + 1)}{1 - 3 x})}$

Factor $- 1$ out of $- 3 (x - 2)$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{- (3 (x - 2)) - (- 3 x + 1)}{1 - 3 x})}$

Factor $- 1$ out of $- (3 (x - 2)) - (- 3 x + 1)$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{- (3 (x - 2) - 3 x + 1)}{1 - 3 x})}$

Apply the distributive property.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{- (3 x + 3 \cdot - 2 - 3 x + 1)}{1 - 3 x})}$

Multiply $3$ by $- 2$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{- (3 x - 6 - 3 x + 1)}{1 - 3 x})}$

Subtract $3 x$ from $3 x$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{- (0 - 6 + 1)}{1 - 3 x})}$

Subtract $6$ from $0$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{- (- 6 + 1)}{1 - 3 x})}$

Add $- 6$ and $1$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{5}{1 - 3 x})}$

Multiply $- 1$ by $- 5$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{5 x}{(1 - 3 x) (\frac{5}{1 - 3 x})}$

Factor $\frac{5}{1 - 3 x}$ out of $\frac{5 x}{(1 - 3 x) \frac{5}{1 - 3 x}}$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{1 - 3 x}{5} \cdot \frac{5 x}{1 - 3 x}$

Cancel the common factor of $1 - 3 x$ .

Cancel the common factor.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{1 - 3 x}{5} \cdot \frac{5 x}{1 - 3 x}$

Rewrite the expression.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{1}{5} \cdot (5 x)$

Cancel the common factor of $5$ .

Factor $5$ out of $5 x$ .

$f (- \frac{x - 2}{1 - 3 x}) = \frac{1}{5} \cdot (5 (x))$

Cancel the common factor.

$f (- \frac{x - 2}{1 - 3 x}) = \frac{1}{5} \cdot (5 x)$

Rewrite the expression.

$f (- \frac{x - 2}{1 - 3 x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x - 2}{1 - 3 x}$ is the inverse of $f (x) = \frac{x - 2}{3 x - 1}$ .

$f^{- 1} (x) = - \frac{x - 2}{1 - 3 x}$

Find the Inverse y=(x-2)/(3x-1)

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