Find the Inverse (9x)/(4x-5)

$\frac{9 x}{4 x - 5}$

Step 1

Interchange the variables.

$x = \frac{9 y}{4 y - 5}$

Step 2

Solve for $y$ .

Multiply the equation by $4 y - 5$ .

$(4 y - 5) (x) = (\frac{9 y}{4 y - 5}) (4 y - 5)$

Simplify the left side.

Apply the distributive property.

$4 y x - 5 x = (\frac{9 y}{4 y - 5}) (4 y - 5)$

Simplify the right side.

Cancel the common factor of $4 y - 5$ .

Cancel the common factor.

$4 y x - 5 x = \frac{9 y}{4 y - 5} (4 y - 5)$

Rewrite the expression.

$4 y x - 5 x = 9 y$

Solve for $y$ .

Subtract $9 y$ from both sides of the equation.

$4 y x - 5 x - 9 y = 0$

Add $5 x$ to both sides of the equation.

$4 y x - 9 y = 5 x$

Factor $y$ out of $4 y x - 9 y$ .

Factor $y$ out of $4 y x$ .

$y (4 x) - 9 y = 5 x$

Factor $y$ out of $- 9 y$ .

$y (4 x) + y \cdot - 9 = 5 x$

Factor $y$ out of $y (4 x) + y \cdot - 9$ .

$y (4 x - 9) = 5 x$

Divide each term in $y (4 x - 9) = 5 x$ by $4 x - 9$ and simplify.

Divide each term in $y (4 x - 9) = 5 x$ by $4 x - 9$ .

$\frac{y (4 x - 9)}{4 x - 9} = \frac{5 x}{4 x - 9}$

Simplify the left side.

Cancel the common factor of $4 x - 9$ .

Cancel the common factor.

$\frac{y (4 x - 9)}{4 x - 9} = \frac{5 x}{4 x - 9}$

Divide $y$ by $1$ .

$y = \frac{5 x}{4 x - 9}$

Step 3

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

$f^{- 1} (x) = \frac{5 x}{4 x - 9}$

Step 4

Verify if $f^{- 1} (x) = \frac{5 x}{4 x - 9}$ is the inverse of $f (x) = \frac{9 x}{4 x - 5}$ .

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 $f^{- 1} (\frac{9 x}{4 x - 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{5 (\frac{9 x}{4 x - 5})}{4 (\frac{9 x}{4 x - 5}) - 9}$

Combine $5$ and $\frac{9 x}{4 x - 5}$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{4 (\frac{9 x}{4 x - 5}) - 9}$

Simplify the denominator.

Multiply $4 \frac{9 x}{4 x - 5}$ .

Combine $4$ and $\frac{9 x}{4 x - 5}$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{4 (9 x)}{4 x - 5} - 9}$

Multiply $9$ by $4$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{36 x}{4 x - 5} - 9}$

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{4 x - 5}{4 x - 5}$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{36 x}{4 x - 5} - 9 \cdot \frac{4 x - 5}{4 x - 5}}$

Combine $- 9$ and $\frac{4 x - 5}{4 x - 5}$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{36 x}{4 x - 5} + \frac{- 9 (4 x - 5)}{4 x - 5}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{36 x - 9 (4 x - 5)}{4 x - 5}}$

Rewrite $\frac{36 x - 9 (4 x - 5)}{4 x - 5}$ in a factored form.

Factor $9$ out of $36 x - 9 (4 x - 5)$ .

Factor $9$ out of $36 x$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{9 (4 x) - 9 (4 x - 5)}{4 x - 5}}$

Factor $9$ out of $- 9 (4 x - 5)$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{9 (4 x) + 9 (- (4 x - 5))}{4 x - 5}}$

Factor $9$ out of $9 (4 x) + 9 (- (4 x - 5))$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{9 (4 x - (4 x - 5))}{4 x - 5}}$

Apply the distributive property.

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{9 (4 x - (4 x) + 5)}{4 x - 5}}$

Multiply $4$ by $- 1$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{9 (4 x - 4 x + 5)}{4 x - 5}}$

Multiply $- 1$ by $- 5$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{9 (4 x - 4 x + 5)}{4 x - 5}}$

Subtract $4 x$ from $4 x$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{9 (0 + 5)}{4 x - 5}}$

Add $0$ and $5$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{9 \cdot 5}{4 x - 5}}$

Multiply $9$ by $5$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{5 (9 x)}{4 x - 5}}{\frac{45}{4 x - 5}}$

Multiply $5$ by $9$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{\frac{45 x}{4 x - 5}}{\frac{45}{4 x - 5}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{45 x}{4 x - 5} \cdot \frac{4 x - 5}{45}$

Cancel the common factor of $45$ .

Factor $45$ out of $45 x$ .

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{45 (x)}{4 x - 5} \cdot \frac{4 x - 5}{45}$

Cancel the common factor.

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{45 x}{4 x - 5} \cdot \frac{4 x - 5}{45}$

Rewrite the expression.

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{x}{4 x - 5} \cdot (4 x - 5)$

Cancel the common factor of $4 x - 5$ .

Cancel the common factor.

$f^{- 1} (\frac{9 x}{4 x - 5}) = \frac{x}{4 x - 5} \cdot (4 x - 5)$

Rewrite the expression.

$f^{- 1} (\frac{9 x}{4 x - 5}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{5 x}{4 x - 9})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x}{4 x - 9}) = \frac{9 (\frac{5 x}{4 x - 9})}{4 (\frac{5 x}{4 x - 9}) - 5}$

Combine $9$ and $\frac{5 x}{4 x - 9}$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{4 (\frac{5 x}{4 x - 9}) - 5}$

Simplify the denominator.

Multiply $4 \frac{5 x}{4 x - 9}$ .

Combine $4$ and $\frac{5 x}{4 x - 9}$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{4 (5 x)}{4 x - 9} - 5}$

Multiply $5$ by $4$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{20 x}{4 x - 9} - 5}$

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{4 x - 9}{4 x - 9}$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{20 x}{4 x - 9} - 5 \cdot \frac{4 x - 9}{4 x - 9}}$

Combine $- 5$ and $\frac{4 x - 9}{4 x - 9}$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{20 x}{4 x - 9} + \frac{- 5 (4 x - 9)}{4 x - 9}}$

Combine the numerators over the common denominator.

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{20 x - 5 (4 x - 9)}{4 x - 9}}$

Rewrite $\frac{20 x - 5 (4 x - 9)}{4 x - 9}$ in a factored form.

Factor $5$ out of $20 x - 5 (4 x - 9)$ .

Factor $5$ out of $20 x$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{5 (4 x) - 5 (4 x - 9)}{4 x - 9}}$

Factor $5$ out of $- 5 (4 x - 9)$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{5 (4 x) + 5 (- (4 x - 9))}{4 x - 9}}$

Factor $5$ out of $5 (4 x) + 5 (- (4 x - 9))$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{5 (4 x - (4 x - 9))}{4 x - 9}}$

Apply the distributive property.

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{5 (4 x - (4 x) + 9)}{4 x - 9}}$

Multiply $4$ by $- 1$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{5 (4 x - 4 x + 9)}{4 x - 9}}$

Multiply $- 1$ by $- 9$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{5 (4 x - 4 x + 9)}{4 x - 9}}$

Subtract $4 x$ from $4 x$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{5 (0 + 9)}{4 x - 9}}$

Add $0$ and $9$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{5 \cdot 9}{4 x - 9}}$

Multiply $5$ by $9$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{9 (5 x)}{4 x - 9}}{\frac{45}{4 x - 9}}$

Multiply $9$ by $5$ .

$f (\frac{5 x}{4 x - 9}) = \frac{\frac{45 x}{4 x - 9}}{\frac{45}{4 x - 9}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5 x}{4 x - 9}) = \frac{45 x}{4 x - 9} \cdot \frac{4 x - 9}{45}$

Cancel the common factor of $45$ .

Factor $45$ out of $45 x$ .

$f (\frac{5 x}{4 x - 9}) = \frac{45 (x)}{4 x - 9} \cdot \frac{4 x - 9}{45}$

Cancel the common factor.

$f (\frac{5 x}{4 x - 9}) = \frac{45 x}{4 x - 9} \cdot \frac{4 x - 9}{45}$

Rewrite the expression.

$f (\frac{5 x}{4 x - 9}) = \frac{x}{4 x - 9} \cdot (4 x - 9)$

Cancel the common factor of $4 x - 9$ .

Cancel the common factor.

$f (\frac{5 x}{4 x - 9}) = \frac{x}{4 x - 9} \cdot (4 x - 9)$

Rewrite the expression.

$f (\frac{5 x}{4 x - 9}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x}{4 x - 9}$ is the inverse of $f (x) = \frac{9 x}{4 x - 5}$ .

$f^{- 1} (x) = \frac{5 x}{4 x - 9}$

Find the Inverse (9x)/(4x-5)

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