Find the Inverse (5x)/(x-9)

$\frac{5 x}{x - 9}$

Step 1

Interchange the variables.

$x = \frac{5 y}{y - 9}$

Step 2

Solve for $y$ .

Multiply the equation by $y - 9$ .

$(y - 9) (x) = (\frac{5 y}{y - 9}) (y - 9)$

Simplify the left side.

Apply the distributive property.

$y x - 9 x = (\frac{5 y}{y - 9}) (y - 9)$

Simplify the right side.

Cancel the common factor of $y - 9$ .

Cancel the common factor.

$y x - 9 x = \frac{5 y}{y - 9} (y - 9)$

Rewrite the expression.

$y x - 9 x = 5 y$

Solve for $y$ .

Subtract $5 y$ from both sides of the equation.

$y x - 9 x - 5 y = 0$

Add $9 x$ to both sides of the equation.

$y x - 5 y = 9 x$

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

Factor $y$ out of $y x$ .

$y (x) - 5 y = 9 x$

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

$y (x) + y \cdot - 5 = 9 x$

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

$y (x - 5) = 9 x$

Divide each term in $y (x - 5) = 9 x$ by $x - 5$ and simplify.

Divide each term in $y (x - 5) = 9 x$ by $x - 5$ .

$\frac{y (x - 5)}{x - 5} = \frac{9 x}{x - 5}$

Simplify the left side.

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{y (x - 5)}{x - 5} = \frac{9 x}{x - 5}$

Divide $y$ by $1$ .

$y = \frac{9 x}{x - 5}$

Step 3

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

$f^{- 1} (x) = \frac{9 x}{x - 5}$

Step 4

Verify if $f^{- 1} (x) = \frac{9 x}{x - 5}$ is the inverse of $f (x) = \frac{5 x}{x - 9}$ .

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 $f^{- 1} (\frac{5 x}{x - 9})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{9 (\frac{5 x}{x - 9})}{(\frac{5 x}{x - 9}) - 5}$

Combine $9$ and $\frac{5 x}{x - 9}$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 x}{x - 9} - 5}$

Simplify the denominator.

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{x - 9}{x - 9}$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 x}{x - 9} - 5 \cdot \frac{x - 9}{x - 9}}$

Combine $- 5$ and $\frac{x - 9}{x - 9}$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 x}{x - 9} + \frac{- 5 (x - 9)}{x - 9}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 x - 5 (x - 9)}{x - 9}}$

Rewrite $\frac{5 x - 5 (x - 9)}{x - 9}$ in a factored form.

Factor $5$ out of $5 x - 5 (x - 9)$ .

Factor $5$ out of $5 x$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 (x) - 5 (x - 9)}{x - 9}}$

Factor $5$ out of $- 5 (x - 9)$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 (x) + 5 (- (x - 9))}{x - 9}}$

Factor $5$ out of $5 (x) + 5 (- (x - 9))$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 (x - (x - 9))}{x - 9}}$

Apply the distributive property.

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 (x - x + 9)}{x - 9}}$

Multiply $- 1$ by $- 9$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 (x - x + 9)}{x - 9}}$

Subtract $x$ from $x$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 (0 + 9)}{x - 9}}$

Add $0$ and $9$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{5 \cdot 9}{x - 9}}$

Multiply $5$ by $9$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{9 (5 x)}{x - 9}}{\frac{45}{x - 9}}$

Multiply $9$ by $5$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{\frac{45 x}{x - 9}}{\frac{45}{x - 9}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{45 x}{x - 9} \cdot \frac{x - 9}{45}$

Cancel the common factor of $45$ .

Factor $45$ out of $45 x$ .

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{45 (x)}{x - 9} \cdot \frac{x - 9}{45}$

Cancel the common factor.

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{45 x}{x - 9} \cdot \frac{x - 9}{45}$

Rewrite the expression.

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{x}{x - 9} \cdot (x - 9)$

Cancel the common factor of $x - 9$ .

Cancel the common factor.

$f^{- 1} (\frac{5 x}{x - 9}) = \frac{x}{x - 9} \cdot (x - 9)$

Rewrite the expression.

$f^{- 1} (\frac{5 x}{x - 9}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{9 x}{x - 5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{9 x}{x - 5}) = \frac{5 (\frac{9 x}{x - 5})}{(\frac{9 x}{x - 5}) - 9}$

Combine $5$ and $\frac{9 x}{x - 5}$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 x}{x - 5} - 9}$

Simplify the denominator.

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 x}{x - 5} - 9 \cdot \frac{x - 5}{x - 5}}$

Combine $- 9$ and $\frac{x - 5}{x - 5}$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 x}{x - 5} + \frac{- 9 (x - 5)}{x - 5}}$

Combine the numerators over the common denominator.

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 x - 9 (x - 5)}{x - 5}}$

Rewrite $\frac{9 x - 9 (x - 5)}{x - 5}$ in a factored form.

Factor $9$ out of $9 x - 9 (x - 5)$ .

Factor $9$ out of $9 x$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 (x) - 9 (x - 5)}{x - 5}}$

Factor $9$ out of $- 9 (x - 5)$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 (x) + 9 (- (x - 5))}{x - 5}}$

Factor $9$ out of $9 (x) + 9 (- (x - 5))$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 (x - (x - 5))}{x - 5}}$

Apply the distributive property.

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 (x - x + 5)}{x - 5}}$

Multiply $- 1$ by $- 5$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 (x - x + 5)}{x - 5}}$

Subtract $x$ from $x$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 (0 + 5)}{x - 5}}$

Add $0$ and $5$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{9 \cdot 5}{x - 5}}$

Multiply $9$ by $5$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{5 (9 x)}{x - 5}}{\frac{45}{x - 5}}$

Multiply $5$ by $9$ .

$f (\frac{9 x}{x - 5}) = \frac{\frac{45 x}{x - 5}}{\frac{45}{x - 5}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{9 x}{x - 5}) = \frac{45 x}{x - 5} \cdot \frac{x - 5}{45}$

Cancel the common factor of $45$ .

Factor $45$ out of $45 x$ .

$f (\frac{9 x}{x - 5}) = \frac{45 (x)}{x - 5} \cdot \frac{x - 5}{45}$

Cancel the common factor.

$f (\frac{9 x}{x - 5}) = \frac{45 x}{x - 5} \cdot \frac{x - 5}{45}$

Rewrite the expression.

$f (\frac{9 x}{x - 5}) = \frac{x}{x - 5} \cdot (x - 5)$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$f (\frac{9 x}{x - 5}) = \frac{x}{x - 5} \cdot (x - 5)$

Rewrite the expression.

$f (\frac{9 x}{x - 5}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{9 x}{x - 5}$ is the inverse of $f (x) = \frac{5 x}{x - 9}$ .

$f^{- 1} (x) = \frac{9 x}{x - 5}$

Find the Inverse (5x)/(x-9)

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