Find the Inverse (3x)/(5x-1)

$\frac{3 x}{5 x - 1}$

Step 1

Interchange the variables.

$x = \frac{3 y}{5 y - 1}$

Step 2

Solve for $y$ .

Multiply the equation by $5 y - 1$ .

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

Simplify the left side.

Simplify $(5 y - 1) (x)$ .

Apply the distributive property.

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

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

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

Simplify the right side.

Cancel the common factor of $5 y - 1$ .

Cancel the common factor.

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

Rewrite the expression.

$5 y x - x = 3 y$

Solve for $y$ .

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

$5 y x - x - 3 y = 0$

Add $x$ to both sides of the equation.

$5 y x - 3 y = x$

Factor $y$ out of $5 y x - 3 y$ .

Factor $y$ out of $5 y x$ .

$y (5 x) - 3 y = x$

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

$y (5 x) + y \cdot - 3 = x$

Factor $y$ out of $y (5 x) + y \cdot - 3$ .

$y (5 x - 3) = x$

Divide each term in $y (5 x - 3) = x$ by $5 x - 3$ and simplify.

Divide each term in $y (5 x - 3) = x$ by $5 x - 3$ .

$\frac{y (5 x - 3)}{5 x - 3} = \frac{x}{5 x - 3}$

Simplify the left side.

Cancel the common factor of $5 x - 3$ .

Cancel the common factor.

$\frac{y (5 x - 3)}{5 x - 3} = \frac{x}{5 x - 3}$

Divide $y$ by $1$ .

$y = \frac{x}{5 x - 3}$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = \frac{x}{5 x - 3}$ is the inverse of $f (x) = \frac{3 x}{5 x - 1}$ .

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 $f^{- 1} (\frac{3 x}{5 x - 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Simplify the denominator.

Multiply $5 \frac{3 x}{5 x - 1}$ .

Combine $5$ and $\frac{3 x}{5 x - 1}$ .

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

Multiply $3$ by $5$ .

$f^{- 1} (\frac{3 x}{5 x - 1}) = \frac{3 x}{5 x - 1} \cdot \frac{1}{\frac{15 x}{5 x - 1} - 3}$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5 x - 1}{5 x - 1}$ .

$f^{- 1} (\frac{3 x}{5 x - 1}) = \frac{3 x}{5 x - 1} \cdot \frac{1}{\frac{15 x}{5 x - 1} - 3 \cdot \frac{5 x - 1}{5 x - 1}}$

Combine $- 3$ and $\frac{5 x - 1}{5 x - 1}$ .

$f^{- 1} (\frac{3 x}{5 x - 1}) = \frac{3 x}{5 x - 1} \cdot \frac{1}{\frac{15 x}{5 x - 1} + \frac{- 3 (5 x - 1)}{5 x - 1}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x}{5 x - 1}) = \frac{3 x}{5 x - 1} \cdot \frac{1}{\frac{15 x - 3 (5 x - 1)}{5 x - 1}}$

Rewrite $\frac{15 x - 3 (5 x - 1)}{5 x - 1}$ in a factored form.

Factor $3$ out of $15 x - 3 (5 x - 1)$ .

Factor $3$ out of $15 x$ .

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

Factor $3$ out of $- 3 (5 x - 1)$ .

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

Factor $3$ out of $3 (5 x) + 3 (- (5 x - 1))$ .

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

Apply the distributive property.

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

Multiply $5$ by $- 1$ .

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

Multiply $- 1$ by $- 1$ .

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

Subtract $5 x$ from $5 x$ .

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

Add $0$ and $1$ .

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

Multiply $3$ by $1$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{5 x - 1}{3}$ by $1$ .

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $5 x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{5 x - 3})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Combine $3$ and $\frac{x}{5 x - 3}$ .

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

Simplify the denominator.

Combine $5$ and $\frac{x}{5 x - 3}$ .

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

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5 x - 3}{5 x - 3}$ .

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

Combine $- 1$ and $\frac{5 x - 3}{5 x - 3}$ .

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{5 x - (5 x - 3)}{5 x - 3}$ in a factored form.

Apply the distributive property.

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

Multiply $5$ by $- 1$ .

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

Multiply $- 1$ by $- 3$ .

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

Subtract $5 x$ from $5 x$ .

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

Add $0$ and $3$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $5 x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{5 x - 3}$ is the inverse of $f (x) = \frac{3 x}{5 x - 1}$ .

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

Find the Inverse (3x)/(5x-1)

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