Find the Inverse (2x)/(x-7)

$\frac{2 x}{x - 7}$

Step 1

Interchange the variables.

$x = \frac{2 y}{y - 7}$

Step 2

Solve for $y$ .

Multiply the equation by $y - 7$ .

$(y - 7) (x) = (\frac{2 y}{y - 7}) (y - 7)$

Simplify the left side.

Apply the distributive property.

$y x - 7 x = (\frac{2 y}{y - 7}) (y - 7)$

Simplify the right side.

Cancel the common factor of $y - 7$ .

Cancel the common factor.

$y x - 7 x = \frac{2 y}{y - 7} (y - 7)$

Rewrite the expression.

$y x - 7 x = 2 y$

Solve for $y$ .

Subtract $2 y$ from both sides of the equation.

$y x - 7 x - 2 y = 0$

Add $7 x$ to both sides of the equation.

$y x - 2 y = 7 x$

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

Factor $y$ out of $y x$ .

$y (x) - 2 y = 7 x$

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

$y (x) + y \cdot - 2 = 7 x$

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

$y (x - 2) = 7 x$

Divide each term in $y (x - 2) = 7 x$ by $x - 2$ and simplify.

Divide each term in $y (x - 2) = 7 x$ by $x - 2$ .

$\frac{y (x - 2)}{x - 2} = \frac{7 x}{x - 2}$

Simplify the left side.

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{y (x - 2)}{x - 2} = \frac{7 x}{x - 2}$

Divide $y$ by $1$ .

$y = \frac{7 x}{x - 2}$

Step 3

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

$f^{- 1} (x) = \frac{7 x}{x - 2}$

Step 4

Verify if $f^{- 1} (x) = \frac{7 x}{x - 2}$ is the inverse of $f (x) = \frac{2 x}{x - 7}$ .

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 $f^{- 1} (\frac{2 x}{x - 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{7 (\frac{2 x}{x - 7})}{(\frac{2 x}{x - 7}) - 2}$

Combine $7$ and $\frac{2 x}{x - 7}$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 x}{x - 7} - 2}$

Simplify the denominator.

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 x}{x - 7} - 2 \cdot \frac{x - 7}{x - 7}}$

Combine $- 2$ and $\frac{x - 7}{x - 7}$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 x}{x - 7} + \frac{- 2 (x - 7)}{x - 7}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 x - 2 (x - 7)}{x - 7}}$

Rewrite $\frac{2 x - 2 (x - 7)}{x - 7}$ in a factored form.

Factor $2$ out of $2 x - 2 (x - 7)$ .

Factor $2$ out of $2 x$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 (x) - 2 (x - 7)}{x - 7}}$

Factor $2$ out of $- 2 (x - 7)$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 (x) + 2 (- (x - 7))}{x - 7}}$

Factor $2$ out of $2 (x) + 2 (- (x - 7))$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 (x - (x - 7))}{x - 7}}$

Apply the distributive property.

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 (x - x + 7)}{x - 7}}$

Multiply $- 1$ by $- 7$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 (x - x + 7)}{x - 7}}$

Subtract $x$ from $x$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 (0 + 7)}{x - 7}}$

Add $0$ and $7$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{2 \cdot 7}{x - 7}}$

Multiply $2$ by $7$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{7 (2 x)}{x - 7}}{\frac{14}{x - 7}}$

Multiply $7$ by $2$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{\frac{14 x}{x - 7}}{\frac{14}{x - 7}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{14 x}{x - 7} \cdot \frac{x - 7}{14}$

Cancel the common factor of $14$ .

Factor $14$ out of $14 x$ .

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{14 (x)}{x - 7} \cdot \frac{x - 7}{14}$

Cancel the common factor.

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{14 x}{x - 7} \cdot \frac{x - 7}{14}$

Rewrite the expression.

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{x}{x - 7} \cdot (x - 7)$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$f^{- 1} (\frac{2 x}{x - 7}) = \frac{x}{x - 7} \cdot (x - 7)$

Rewrite the expression.

$f^{- 1} (\frac{2 x}{x - 7}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{7 x}{x - 2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7 x}{x - 2}) = \frac{2 (\frac{7 x}{x - 2})}{(\frac{7 x}{x - 2}) - 7}$

Combine $2$ and $\frac{7 x}{x - 2}$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 x}{x - 2} - 7}$

Simplify the denominator.

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 x}{x - 2} - 7 \cdot \frac{x - 2}{x - 2}}$

Combine $- 7$ and $\frac{x - 2}{x - 2}$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 x}{x - 2} + \frac{- 7 (x - 2)}{x - 2}}$

Combine the numerators over the common denominator.

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 x - 7 (x - 2)}{x - 2}}$

Rewrite $\frac{7 x - 7 (x - 2)}{x - 2}$ in a factored form.

Factor $7$ out of $7 x - 7 (x - 2)$ .

Factor $7$ out of $7 x$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 (x) - 7 (x - 2)}{x - 2}}$

Factor $7$ out of $- 7 (x - 2)$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 (x) + 7 (- (x - 2))}{x - 2}}$

Factor $7$ out of $7 (x) + 7 (- (x - 2))$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 (x - (x - 2))}{x - 2}}$

Apply the distributive property.

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 (x - x + 2)}{x - 2}}$

Multiply $- 1$ by $- 2$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 (x - x + 2)}{x - 2}}$

Subtract $x$ from $x$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 (0 + 2)}{x - 2}}$

Add $0$ and $2$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{7 \cdot 2}{x - 2}}$

Multiply $7$ by $2$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{2 (7 x)}{x - 2}}{\frac{14}{x - 2}}$

Multiply $2$ by $7$ .

$f (\frac{7 x}{x - 2}) = \frac{\frac{14 x}{x - 2}}{\frac{14}{x - 2}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 x}{x - 2}) = \frac{14 x}{x - 2} \cdot \frac{x - 2}{14}$

Cancel the common factor of $14$ .

Factor $14$ out of $14 x$ .

$f (\frac{7 x}{x - 2}) = \frac{14 (x)}{x - 2} \cdot \frac{x - 2}{14}$

Cancel the common factor.

$f (\frac{7 x}{x - 2}) = \frac{14 x}{x - 2} \cdot \frac{x - 2}{14}$

Rewrite the expression.

$f (\frac{7 x}{x - 2}) = \frac{x}{x - 2} \cdot (x - 2)$

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$f (\frac{7 x}{x - 2}) = \frac{x}{x - 2} \cdot (x - 2)$

Rewrite the expression.

$f (\frac{7 x}{x - 2}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x}{x - 2}$ is the inverse of $f (x) = \frac{2 x}{x - 7}$ .

$f^{- 1} (x) = \frac{7 x}{x - 2}$

Find the Inverse (2x)/(x-7)

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