Find the Inverse (3x-4)/(5-x)

$\frac{3 x - 4}{5 - x}$

Step 1

Interchange the variables.

$x = \frac{3 y - 4}{5 - y}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{3 y - 4}{5 - y} = x$ .

$\frac{3 y - 4}{5 - y} = x$

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.

$5 - y, 1$

Remove parentheses.

$5 - y, 1$

The LCM of one and any expression is the expression.

$5 - y$

Multiply each term in $\frac{3 y - 4}{5 - y} = x$ by $5 - y$ to eliminate the fractions.

Multiply each term in $\frac{3 y - 4}{5 - y} = x$ by $5 - y$ .

$\frac{3 y - 4}{5 - y} (5 - y) = x (5 - y)$

Simplify the left side.

Cancel the common factor of $5 - y$ .

Cancel the common factor.

$\frac{3 y - 4}{5 - y} (5 - y) = x (5 - y)$

Rewrite the expression.

$3 y - 4 = x (5 - y)$

Simplify the right side.

Apply the distributive property.

$3 y - 4 = x \cdot 5 + x (- y)$

Reorder.

Move $5$ to the left of $x$ .

$3 y - 4 = 5 \cdot x + x (- y)$

Rewrite using the commutative property of multiplication.

$3 y - 4 = 5 x - x y$

Solve the equation.

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

$3 y - 4 + x y = 5 x$

Add $4$ to both sides of the equation.

$3 y + x y = 5 x + 4$

Factor $y$ out of $3 y + x y$ .

Factor $y$ out of $3 y$ .

$y \cdot 3 + x y = 5 x + 4$

Factor $y$ out of $x y$ .

$y \cdot 3 + y x = 5 x + 4$

Factor $y$ out of $y \cdot 3 + y x$ .

$y (3 + x) = 5 x + 4$

Divide each term in $y (3 + x) = 5 x + 4$ by $3 + x$ and simplify.

Divide each term in $y (3 + x) = 5 x + 4$ by $3 + x$ .

$\frac{y (3 + x)}{3 + x} = \frac{5 x}{3 + x} + \frac{4}{3 + x}$

Simplify the left side.

Cancel the common factor of $3 + x$ .

Cancel the common factor.

$\frac{y (3 + x)}{3 + x} = \frac{5 x}{3 + x} + \frac{4}{3 + x}$

Divide $y$ by $1$ .

$y = \frac{5 x}{3 + x} + \frac{4}{3 + x}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{5 x + 4}{3 + x}$

Step 3

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

$f^{- 1} (x) = \frac{5 x + 4}{3 + x}$

Step 4

Verify if $f^{- 1} (x) = \frac{5 x + 4}{3 + x}$ is the inverse of $f (x) = \frac{3 x - 4}{5 - x}$ .

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 $f^{- 1} (\frac{3 x - 4}{5 - x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{5 (\frac{3 x - 4}{5 - x}) + 4}{3 + \frac{3 x - 4}{5 - x}}$

Remove parentheses.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{5 (\frac{3 x - 4}{5 - x}) + 4}{3 + \frac{3 x - 4}{5 - x}}$

Multiply the numerator and denominator of the complex fraction by $5 - x$ .

Multiply $\frac{5 \frac{3 x - 4}{5 - x} + 4}{3 + \frac{3 x - 4}{5 - x}}$ by $\frac{5 - x}{5 - x}$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{5 - x}{5 - x} \cdot \frac{5 (\frac{3 x - 4}{5 - x}) + 4}{3 + \frac{3 x - 4}{5 - x}}$

Combine.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{3 x - 4}{5 - x}) + 4)}{(5 - x) (3 + \frac{3 x - 4}{5 - x})}$

Apply the distributive property.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{3 x - 4}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 3 + (5 - x) (\frac{3 x - 4}{5 - x})}$

Cancel the common factor of $5 - x$ .

Cancel the common factor.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{3 x - 4}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 3 + (5 - x) (\frac{3 x - 4}{5 - x})}$

Rewrite the expression.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{3 x - 4}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 3 + 3 x - 4}$

Simplify the numerator.

Factor $5 - x$ out of $(5 - x) \cdot 5 \frac{3 x - 4}{5 - x} + (5 - x) \cdot 4$ .

Factor $5 - x$ out of $(5 - x) \cdot 5 \frac{3 x - 4}{5 - x}$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{3 x - 4}{5 - x})) + (5 - x) \cdot 4}{(5 - x) \cdot 3 + 3 x - 4}$

Factor $5 - x$ out of $(5 - x) \cdot 4$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{3 x - 4}{5 - x})) + (5 - x) (4)}{(5 - x) \cdot 3 + 3 x - 4}$

Factor $5 - x$ out of $(5 - x) (5 \frac{3 x - 4}{5 - x}) + (5 - x) (4)$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (5 (\frac{3 x - 4}{5 - x}) + 4)}{(5 - x) \cdot 3 + 3 x - 4}$

Combine $5$ and $\frac{3 x - 4}{5 - x}$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{5 (3 x - 4)}{5 - x} + 4)}{(5 - x) \cdot 3 + 3 x - 4}$

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

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{5 (3 x - 4)}{5 - x} + \frac{4 (5 - x)}{5 - x})}{(5 - x) \cdot 3 + 3 x - 4}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{5 (3 x - 4) + 4 (5 - x)}{5 - x})}{(5 - x) \cdot 3 + 3 x - 4}$

Reorder terms.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{5 (3 x - 4) + 4 (5 - x)}{- x + 5})}{(5 - x) \cdot 3 + 3 x - 4}$

Rewrite $\frac{5 (3 x - 4) + 4 (5 - x)}{- x + 5}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{5 (3 x) + 5 \cdot - 4 + 4 (5 - x)}{- x + 5})}{(5 - x) \cdot 3 + 3 x - 4}$

Multiply $3$ by $5$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{15 x + 5 \cdot - 4 + 4 (5 - x)}{- x + 5})}{(5 - x) \cdot 3 + 3 x - 4}$

Multiply $5$ by $- 4$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{15 x - 20 + 4 (5 - x)}{- x + 5})}{(5 - x) \cdot 3 + 3 x - 4}$

Apply the distributive property.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{15 x - 20 + 4 \cdot 5 + 4 (- x)}{- x + 5})}{(5 - x) \cdot 3 + 3 x - 4}$

Multiply $4$ by $5$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{15 x - 20 + 20 + 4 (- x)}{- x + 5})}{(5 - x) \cdot 3 + 3 x - 4}$

Multiply $- 1$ by $4$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{15 x - 20 + 20 - 4 x}{- x + 5})}{(5 - x) \cdot 3 + 3 x - 4}$

Subtract $4 x$ from $15 x$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{11 x - 20 + 20}{- x + 5})}{(5 - x) \cdot 3 + 3 x - 4}$

Add $- 20$ and $20$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{11 x + 0}{- x + 5})}{(5 - x) \cdot 3 + 3 x - 4}$

Add $11 x$ and $0$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{11 x}{- x + 5})}{(5 - x) \cdot 3 + 3 x - 4}$

Simplify the denominator.

Apply the distributive property.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{11 x}{- x + 5})}{5 \cdot 3 - x \cdot 3 + 3 x - 4}$

Multiply $5$ by $3$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{11 x}{- x + 5})}{15 - x \cdot 3 + 3 x - 4}$

Multiply $3$ by $- 1$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{11 x}{- x + 5})}{15 - 3 x + 3 x - 4}$

Subtract $4$ from $15$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{11 x}{- x + 5})}{- 3 x + 3 x + 11}$

Add $- 3 x$ and $3 x$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{11 x}{- x + 5})}{0 + 11}$

Add $0$ and $11$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{(5 - x) (\frac{11 x}{- x + 5})}{11}$

Simplify terms.

Factor $\frac{11 x}{- x + 5}$ out of $\frac{(5 - x) \frac{11 x}{- x + 5}}{11}$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{11 x}{- x + 5} \cdot \frac{5 - x}{11}$

Cancel the common factor of $11$ .

Factor $11$ out of $11 x$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{11 (x)}{- x + 5} \cdot \frac{5 - x}{11}$

Cancel the common factor.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{11 x}{- x + 5} \cdot \frac{5 - x}{11}$

Rewrite the expression.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{x}{- x + 5} \cdot (5 - x)$

Multiply $\frac{x}{- x + 5}$ by $5 - x$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{x (5 - x)}{- x + 5}$

Cancel the common factor of $5 - x$ and $- x + 5$ .

Reorder terms.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{x (- x + 5)}{- x + 5}$

Cancel the common factor.

$f^{- 1} (\frac{3 x - 4}{5 - x}) = \frac{x (- x + 5)}{- x + 5}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{3 x - 4}{5 - x}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{5 x + 4}{3 + x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{3 (\frac{5 x + 4}{3 + x}) - 4}{5 - (\frac{5 x + 4}{3 + x})}$

Multiply the numerator and denominator of the complex fraction by $3 + x$ .

Multiply $\frac{3 \frac{5 x + 4}{3 + x} - 4}{5 - \frac{5 x + 4}{3 + x}}$ by $\frac{3 + x}{3 + x}$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{3 + x}{3 + x} \cdot \frac{3 (\frac{5 x + 4}{3 + x}) - 4}{5 - \frac{5 x + 4}{3 + x}}$

Combine.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (3 (\frac{5 x + 4}{3 + x}) - 4)}{(3 + x) (5 - \frac{5 x + 4}{3 + x})}$

Apply the distributive property.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (3 (\frac{5 x + 4}{3 + x})) + (3 + x) \cdot - 4}{(3 + x) \cdot 5 + (3 + x) (- \frac{5 x + 4}{3 + x})}$

Cancel the common factor of $3 + x$ .

Move the leading negative in $- \frac{5 x + 4}{3 + x}$ into the numerator.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (3 (\frac{5 x + 4}{3 + x})) + (3 + x) \cdot - 4}{(3 + x) \cdot 5 + (3 + x) (\frac{- (5 x + 4)}{3 + x})}$

Cancel the common factor.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (3 (\frac{5 x + 4}{3 + x})) + (3 + x) \cdot - 4}{(3 + x) \cdot 5 + (3 + x) (\frac{- (5 x + 4)}{3 + x})}$

Rewrite the expression.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (3 (\frac{5 x + 4}{3 + x})) + (3 + x) \cdot - 4}{(3 + x) \cdot 5 - (5 x + 4)}$

Simplify the numerator.

Factor $3 + x$ out of $(3 + x) \cdot 3 \frac{5 x + 4}{3 + x} + (3 + x) \cdot - 4$ .

Factor $3 + x$ out of $(3 + x) \cdot 3 \frac{5 x + 4}{3 + x}$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (3 (\frac{5 x + 4}{3 + x})) + (3 + x) \cdot - 4}{(3 + x) \cdot 5 - (5 x + 4)}$

Factor $3 + x$ out of $(3 + x) \cdot - 4$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (3 (\frac{5 x + 4}{3 + x})) + (3 + x) (- 4)}{(3 + x) \cdot 5 - (5 x + 4)}$

Factor $3 + x$ out of $(3 + x) (3 \frac{5 x + 4}{3 + x}) + (3 + x) (- 4)$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (3 (\frac{5 x + 4}{3 + x}) - 4)}{(3 + x) \cdot 5 - (5 x + 4)}$

Combine $3$ and $\frac{5 x + 4}{3 + x}$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{3 (5 x + 4)}{3 + x} - 4)}{(3 + x) \cdot 5 - (5 x + 4)}$

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3 + x}{3 + x}$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{3 (5 x + 4)}{3 + x} - 4 \cdot \frac{3 + x}{3 + x})}{(3 + x) \cdot 5 - (5 x + 4)}$

Combine $- 4$ and $\frac{3 + x}{3 + x}$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{3 (5 x + 4)}{3 + x} + \frac{- 4 (3 + x)}{3 + x})}{(3 + x) \cdot 5 - (5 x + 4)}$

Combine the numerators over the common denominator.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{3 (5 x + 4) - 4 (3 + x)}{3 + x})}{(3 + x) \cdot 5 - (5 x + 4)}$

Reorder terms.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{3 (5 x + 4) - 4 (3 + x)}{x + 3})}{(3 + x) \cdot 5 - (5 x + 4)}$

Rewrite $\frac{3 (5 x + 4) - 4 (3 + x)}{x + 3}$ in a factored form.

Apply the distributive property.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{3 (5 x) + 3 \cdot 4 - 4 (3 + x)}{x + 3})}{(3 + x) \cdot 5 - (5 x + 4)}$

Multiply $5$ by $3$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{15 x + 3 \cdot 4 - 4 (3 + x)}{x + 3})}{(3 + x) \cdot 5 - (5 x + 4)}$

Multiply $3$ by $4$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{15 x + 12 - 4 (3 + x)}{x + 3})}{(3 + x) \cdot 5 - (5 x + 4)}$

Apply the distributive property.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{15 x + 12 - 4 \cdot 3 - 4 x}{x + 3})}{(3 + x) \cdot 5 - (5 x + 4)}$

Multiply $- 4$ by $3$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{15 x + 12 - 12 - 4 x}{x + 3})}{(3 + x) \cdot 5 - (5 x + 4)}$

Subtract $4 x$ from $15 x$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x + 12 - 12}{x + 3})}{(3 + x) \cdot 5 - (5 x + 4)}$

Subtract $12$ from $12$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x + 0}{x + 3})}{(3 + x) \cdot 5 - (5 x + 4)}$

Add $11 x$ and $0$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x}{x + 3})}{(3 + x) \cdot 5 - (5 x + 4)}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x}{x + 3})}{3 \cdot 5 + x \cdot 5 - (5 x + 4)}$

Multiply $3$ by $5$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x}{x + 3})}{15 + x \cdot 5 - (5 x + 4)}$

Move $5$ to the left of $x$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x}{x + 3})}{15 + 5 \cdot x - (5 x + 4)}$

Apply the distributive property.

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x}{x + 3})}{15 + 5 x - (5 x) - 1 \cdot 4}$

Multiply $5$ by $- 1$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x}{x + 3})}{15 + 5 x - 5 x - 1 \cdot 4}$

Multiply $- 1$ by $4$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x}{x + 3})}{15 + 5 x - 5 x - 4}$

Subtract $4$ from $15$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x}{x + 3})}{5 x - 5 x + 11}$

Subtract $5 x$ from $5 x$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x}{x + 3})}{0 + 11}$

Add $0$ and $11$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{(3 + x) (\frac{11 x}{x + 3})}{11}$

Simplify terms.

Factor $\frac{11 x}{x + 3}$ out of $\frac{(3 + x) \frac{11 x}{x + 3}}{11}$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{11 x}{x + 3} \cdot \frac{3 + x}{11}$

Cancel the common factor of $11$ .

Factor $11$ out of $11 x$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{11 (x)}{x + 3} \cdot \frac{3 + x}{11}$

Cancel the common factor.

$f (\frac{5 x + 4}{3 + x}) = \frac{11 x}{x + 3} \cdot \frac{3 + x}{11}$

Rewrite the expression.

$f (\frac{5 x + 4}{3 + x}) = \frac{x}{x + 3} \cdot (3 + x)$

Multiply $\frac{x}{x + 3}$ by $3 + x$ .

$f (\frac{5 x + 4}{3 + x}) = \frac{x (3 + x)}{x + 3}$

Cancel the common factor of $3 + x$ and $x + 3$ .

Reorder terms.

$f (\frac{5 x + 4}{3 + x}) = \frac{x (x + 3)}{x + 3}$

Cancel the common factor.

$f (\frac{5 x + 4}{3 + x}) = \frac{x (x + 3)}{x + 3}$

Divide $x$ by $1$ .

$f (\frac{5 x + 4}{3 + x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x + 4}{3 + x}$ is the inverse of $f (x) = \frac{3 x - 4}{5 - x}$ .

$f^{- 1} (x) = \frac{5 x + 4}{3 + x}$

Find the Inverse (3x-4)/(5-x)

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