Find the Inverse (9x-4)/(7-x)

$\frac{9 x - 4}{7 - x}$

Step 1

Interchange the variables.

$x = \frac{9 y - 4}{7 - y}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{9 y - 4}{7 - y} = x$ .

$\frac{9 y - 4}{7 - y} = x$

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.

$7 - y, 1$

Remove parentheses.

$7 - y, 1$

The LCM of one and any expression is the expression.

$7 - y$

Multiply each term in $\frac{9 y - 4}{7 - y} = x$ by $7 - y$ to eliminate the fractions.

Multiply each term in $\frac{9 y - 4}{7 - y} = x$ by $7 - y$ .

$\frac{9 y - 4}{7 - y} (7 - y) = x (7 - y)$

Simplify the left side.

Cancel the common factor of $7 - y$ .

Cancel the common factor.

$\frac{9 y - 4}{7 - y} (7 - y) = x (7 - y)$

Rewrite the expression.

$9 y - 4 = x (7 - y)$

Simplify the right side.

Apply the distributive property.

$9 y - 4 = x \cdot 7 + x (- y)$

Reorder.

Move $7$ to the left of $x$ .

$9 y - 4 = 7 \cdot x + x (- y)$

Rewrite using the commutative property of multiplication.

$9 y - 4 = 7 x - x y$

Solve the equation.

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

$9 y - 4 + x y = 7 x$

Add $4$ to both sides of the equation.

$9 y + x y = 7 x + 4$

Factor $y$ out of $9 y + x y$ .

Factor $y$ out of $9 y$ .

$y \cdot 9 + x y = 7 x + 4$

Factor $y$ out of $x y$ .

$y \cdot 9 + y x = 7 x + 4$

Factor $y$ out of $y \cdot 9 + y x$ .

$y (9 + x) = 7 x + 4$

Divide each term in $y (9 + x) = 7 x + 4$ by $9 + x$ and simplify.

Divide each term in $y (9 + x) = 7 x + 4$ by $9 + x$ .

$\frac{y (9 + x)}{9 + x} = \frac{7 x}{9 + x} + \frac{4}{9 + x}$

Simplify the left side.

Cancel the common factor of $9 + x$ .

Cancel the common factor.

$\frac{y (9 + x)}{9 + x} = \frac{7 x}{9 + x} + \frac{4}{9 + x}$

Divide $y$ by $1$ .

$y = \frac{7 x}{9 + x} + \frac{4}{9 + x}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{7 x + 4}{9 + x}$

Step 3

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

$f^{- 1} (x) = \frac{7 x + 4}{9 + x}$

Step 4

Verify if $f^{- 1} (x) = \frac{7 x + 4}{9 + x}$ is the inverse of $f (x) = \frac{9 x - 4}{7 - x}$ .

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 $f^{- 1} (\frac{9 x - 4}{7 - x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{7 (\frac{9 x - 4}{7 - x}) + 4}{9 + \frac{9 x - 4}{7 - x}}$

Remove parentheses.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{7 (\frac{9 x - 4}{7 - x}) + 4}{9 + \frac{9 x - 4}{7 - x}}$

Multiply the numerator and denominator of the complex fraction by $7 - x$ .

Multiply $\frac{7 \frac{9 x - 4}{7 - x} + 4}{9 + \frac{9 x - 4}{7 - x}}$ by $\frac{7 - x}{7 - x}$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{7 - x}{7 - x} \cdot \frac{7 (\frac{9 x - 4}{7 - x}) + 4}{9 + \frac{9 x - 4}{7 - x}}$

Combine.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (7 (\frac{9 x - 4}{7 - x}) + 4)}{(7 - x) (9 + \frac{9 x - 4}{7 - x})}$

Apply the distributive property.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (7 (\frac{9 x - 4}{7 - x})) + (7 - x) \cdot 4}{(7 - x) \cdot 9 + (7 - x) (\frac{9 x - 4}{7 - x})}$

Cancel the common factor of $7 - x$ .

Cancel the common factor.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (7 (\frac{9 x - 4}{7 - x})) + (7 - x) \cdot 4}{(7 - x) \cdot 9 + (7 - x) (\frac{9 x - 4}{7 - x})}$

Rewrite the expression.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (7 (\frac{9 x - 4}{7 - x})) + (7 - x) \cdot 4}{(7 - x) \cdot 9 + 9 x - 4}$

Simplify the numerator.

Factor $7 - x$ out of $(7 - x) \cdot 7 \frac{9 x - 4}{7 - x} + (7 - x) \cdot 4$ .

Factor $7 - x$ out of $(7 - x) \cdot 7 \frac{9 x - 4}{7 - x}$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (7 (\frac{9 x - 4}{7 - x})) + (7 - x) \cdot 4}{(7 - x) \cdot 9 + 9 x - 4}$

Factor $7 - x$ out of $(7 - x) \cdot 4$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (7 (\frac{9 x - 4}{7 - x})) + (7 - x) (4)}{(7 - x) \cdot 9 + 9 x - 4}$

Factor $7 - x$ out of $(7 - x) (7 \frac{9 x - 4}{7 - x}) + (7 - x) (4)$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (7 (\frac{9 x - 4}{7 - x}) + 4)}{(7 - x) \cdot 9 + 9 x - 4}$

Combine $7$ and $\frac{9 x - 4}{7 - x}$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{7 (9 x - 4)}{7 - x} + 4)}{(7 - x) \cdot 9 + 9 x - 4}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{7 - x}{7 - x}$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{7 (9 x - 4)}{7 - x} + \frac{4 (7 - x)}{7 - x})}{(7 - x) \cdot 9 + 9 x - 4}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{7 (9 x - 4) + 4 (7 - x)}{7 - x})}{(7 - x) \cdot 9 + 9 x - 4}$

Reorder terms.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{7 (9 x - 4) + 4 (7 - x)}{- x + 7})}{(7 - x) \cdot 9 + 9 x - 4}$

Rewrite $\frac{7 (9 x - 4) + 4 (7 - x)}{- x + 7}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{7 (9 x) + 7 \cdot - 4 + 4 (7 - x)}{- x + 7})}{(7 - x) \cdot 9 + 9 x - 4}$

Multiply $9$ by $7$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{63 x + 7 \cdot - 4 + 4 (7 - x)}{- x + 7})}{(7 - x) \cdot 9 + 9 x - 4}$

Multiply $7$ by $- 4$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{63 x - 28 + 4 (7 - x)}{- x + 7})}{(7 - x) \cdot 9 + 9 x - 4}$

Apply the distributive property.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{63 x - 28 + 4 \cdot 7 + 4 (- x)}{- x + 7})}{(7 - x) \cdot 9 + 9 x - 4}$

Multiply $4$ by $7$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{63 x - 28 + 28 + 4 (- x)}{- x + 7})}{(7 - x) \cdot 9 + 9 x - 4}$

Multiply $- 1$ by $4$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{63 x - 28 + 28 - 4 x}{- x + 7})}{(7 - x) \cdot 9 + 9 x - 4}$

Subtract $4 x$ from $63 x$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{59 x - 28 + 28}{- x + 7})}{(7 - x) \cdot 9 + 9 x - 4}$

Add $- 28$ and $28$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{59 x + 0}{- x + 7})}{(7 - x) \cdot 9 + 9 x - 4}$

Add $59 x$ and $0$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{59 x}{- x + 7})}{(7 - x) \cdot 9 + 9 x - 4}$

Simplify the denominator.

Apply the distributive property.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{59 x}{- x + 7})}{7 \cdot 9 - x \cdot 9 + 9 x - 4}$

Multiply $7$ by $9$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{59 x}{- x + 7})}{63 - x \cdot 9 + 9 x - 4}$

Multiply $9$ by $- 1$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{59 x}{- x + 7})}{63 - 9 x + 9 x - 4}$

Subtract $4$ from $63$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{59 x}{- x + 7})}{- 9 x + 9 x + 59}$

Add $- 9 x$ and $9 x$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{59 x}{- x + 7})}{0 + 59}$

Add $0$ and $59$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{(7 - x) (\frac{59 x}{- x + 7})}{59}$

Simplify terms.

Factor $\frac{59 x}{- x + 7}$ out of $\frac{(7 - x) \frac{59 x}{- x + 7}}{59}$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{59 x}{- x + 7} \cdot \frac{7 - x}{59}$

Cancel the common factor of $59$ .

Factor $59$ out of $59 x$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{59 (x)}{- x + 7} \cdot \frac{7 - x}{59}$

Cancel the common factor.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{59 x}{- x + 7} \cdot \frac{7 - x}{59}$

Rewrite the expression.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{x}{- x + 7} \cdot (7 - x)$

Multiply $\frac{x}{- x + 7}$ by $7 - x$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{x (7 - x)}{- x + 7}$

Cancel the common factor of $7 - x$ and $- x + 7$ .

Reorder terms.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{x (- x + 7)}{- x + 7}$

Cancel the common factor.

$f^{- 1} (\frac{9 x - 4}{7 - x}) = \frac{x (- x + 7)}{- x + 7}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{9 x - 4}{7 - x}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{7 x + 4}{9 + x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{9 (\frac{7 x + 4}{9 + x}) - 4}{7 - (\frac{7 x + 4}{9 + x})}$

Multiply the numerator and denominator of the complex fraction by $9 + x$ .

Multiply $\frac{9 \frac{7 x + 4}{9 + x} - 4}{7 - \frac{7 x + 4}{9 + x}}$ by $\frac{9 + x}{9 + x}$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{9 + x}{9 + x} \cdot \frac{9 (\frac{7 x + 4}{9 + x}) - 4}{7 - \frac{7 x + 4}{9 + x}}$

Combine.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (9 (\frac{7 x + 4}{9 + x}) - 4)}{(9 + x) (7 - \frac{7 x + 4}{9 + x})}$

Apply the distributive property.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (9 (\frac{7 x + 4}{9 + x})) + (9 + x) \cdot - 4}{(9 + x) \cdot 7 + (9 + x) (- \frac{7 x + 4}{9 + x})}$

Cancel the common factor of $9 + x$ .

Move the leading negative in $- \frac{7 x + 4}{9 + x}$ into the numerator.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (9 (\frac{7 x + 4}{9 + x})) + (9 + x) \cdot - 4}{(9 + x) \cdot 7 + (9 + x) (\frac{- (7 x + 4)}{9 + x})}$

Cancel the common factor.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (9 (\frac{7 x + 4}{9 + x})) + (9 + x) \cdot - 4}{(9 + x) \cdot 7 + (9 + x) (\frac{- (7 x + 4)}{9 + x})}$

Rewrite the expression.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (9 (\frac{7 x + 4}{9 + x})) + (9 + x) \cdot - 4}{(9 + x) \cdot 7 - (7 x + 4)}$

Simplify the numerator.

Factor $9 + x$ out of $(9 + x) \cdot 9 \frac{7 x + 4}{9 + x} + (9 + x) \cdot - 4$ .

Factor $9 + x$ out of $(9 + x) \cdot 9 \frac{7 x + 4}{9 + x}$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (9 (\frac{7 x + 4}{9 + x})) + (9 + x) \cdot - 4}{(9 + x) \cdot 7 - (7 x + 4)}$

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

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (9 (\frac{7 x + 4}{9 + x})) + (9 + x) (- 4)}{(9 + x) \cdot 7 - (7 x + 4)}$

Factor $9 + x$ out of $(9 + x) (9 \frac{7 x + 4}{9 + x}) + (9 + x) (- 4)$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (9 (\frac{7 x + 4}{9 + x}) - 4)}{(9 + x) \cdot 7 - (7 x + 4)}$

Combine $9$ and $\frac{7 x + 4}{9 + x}$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{9 (7 x + 4)}{9 + x} - 4)}{(9 + x) \cdot 7 - (7 x + 4)}$

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

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{9 (7 x + 4)}{9 + x} - 4 \cdot \frac{9 + x}{9 + x})}{(9 + x) \cdot 7 - (7 x + 4)}$

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

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{9 (7 x + 4)}{9 + x} + \frac{- 4 (9 + x)}{9 + x})}{(9 + x) \cdot 7 - (7 x + 4)}$

Combine the numerators over the common denominator.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{9 (7 x + 4) - 4 (9 + x)}{9 + x})}{(9 + x) \cdot 7 - (7 x + 4)}$

Reorder terms.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{9 (7 x + 4) - 4 (9 + x)}{x + 9})}{(9 + x) \cdot 7 - (7 x + 4)}$

Rewrite $\frac{9 (7 x + 4) - 4 (9 + x)}{x + 9}$ in a factored form.

Apply the distributive property.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{9 (7 x) + 9 \cdot 4 - 4 (9 + x)}{x + 9})}{(9 + x) \cdot 7 - (7 x + 4)}$

Multiply $7$ by $9$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{63 x + 9 \cdot 4 - 4 (9 + x)}{x + 9})}{(9 + x) \cdot 7 - (7 x + 4)}$

Multiply $9$ by $4$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{63 x + 36 - 4 (9 + x)}{x + 9})}{(9 + x) \cdot 7 - (7 x + 4)}$

Apply the distributive property.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{63 x + 36 - 4 \cdot 9 - 4 x}{x + 9})}{(9 + x) \cdot 7 - (7 x + 4)}$

Multiply $- 4$ by $9$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{63 x + 36 - 36 - 4 x}{x + 9})}{(9 + x) \cdot 7 - (7 x + 4)}$

Subtract $4 x$ from $63 x$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x + 36 - 36}{x + 9})}{(9 + x) \cdot 7 - (7 x + 4)}$

Subtract $36$ from $36$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x + 0}{x + 9})}{(9 + x) \cdot 7 - (7 x + 4)}$

Add $59 x$ and $0$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x}{x + 9})}{(9 + x) \cdot 7 - (7 x + 4)}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x}{x + 9})}{9 \cdot 7 + x \cdot 7 - (7 x + 4)}$

Multiply $9$ by $7$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x}{x + 9})}{63 + x \cdot 7 - (7 x + 4)}$

Move $7$ to the left of $x$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x}{x + 9})}{63 + 7 \cdot x - (7 x + 4)}$

Apply the distributive property.

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x}{x + 9})}{63 + 7 x - (7 x) - 1 \cdot 4}$

Multiply $7$ by $- 1$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x}{x + 9})}{63 + 7 x - 7 x - 1 \cdot 4}$

Multiply $- 1$ by $4$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x}{x + 9})}{63 + 7 x - 7 x - 4}$

Subtract $4$ from $63$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x}{x + 9})}{7 x - 7 x + 59}$

Subtract $7 x$ from $7 x$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x}{x + 9})}{0 + 59}$

Add $0$ and $59$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{(9 + x) (\frac{59 x}{x + 9})}{59}$

Simplify terms.

Factor $\frac{59 x}{x + 9}$ out of $\frac{(9 + x) \frac{59 x}{x + 9}}{59}$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{59 x}{x + 9} \cdot \frac{9 + x}{59}$

Cancel the common factor of $59$ .

Factor $59$ out of $59 x$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{59 (x)}{x + 9} \cdot \frac{9 + x}{59}$

Cancel the common factor.

$f (\frac{7 x + 4}{9 + x}) = \frac{59 x}{x + 9} \cdot \frac{9 + x}{59}$

Rewrite the expression.

$f (\frac{7 x + 4}{9 + x}) = \frac{x}{x + 9} \cdot (9 + x)$

Multiply $\frac{x}{x + 9}$ by $9 + x$ .

$f (\frac{7 x + 4}{9 + x}) = \frac{x (9 + x)}{x + 9}$

Cancel the common factor of $9 + x$ and $x + 9$ .

Reorder terms.

$f (\frac{7 x + 4}{9 + x}) = \frac{x (x + 9)}{x + 9}$

Cancel the common factor.

$f (\frac{7 x + 4}{9 + x}) = \frac{x (x + 9)}{x + 9}$

Divide $x$ by $1$ .

$f (\frac{7 x + 4}{9 + x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x + 4}{9 + x}$ is the inverse of $f (x) = \frac{9 x - 4}{7 - x}$ .

$f^{- 1} (x) = \frac{7 x + 4}{9 + x}$

Find the Inverse (9x-4)/(7-x)

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