Find the Inverse (3x-8)/(7-x)

$\frac{3 x - 8}{7 - x}$

Step 1

Interchange the variables.

$x = \frac{3 y - 8}{7 - y}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{3 y - 8}{7 - y} = x$ .

$\frac{3 y - 8}{7 - y} = x$

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 $\frac{3 y - 8}{7 - y} = x$ by $7 - y$ to eliminate the fractions.

Multiply each term in $\frac{3 y - 8}{7 - y} = x$ by $7 - y$ .

$\frac{3 y - 8}{7 - y} (7 - y) = x (7 - y)$

Simplify the left side.

Cancel the common factor of $7 - y$ .

Cancel the common factor.

$\frac{3 y - 8}{7 - y} (7 - y) = x (7 - y)$

Rewrite the expression.

$3 y - 8 = x (7 - y)$

Simplify the right side.

Apply the distributive property.

$3 y - 8 = x \cdot 7 + x (- y)$

Reorder.

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

$3 y - 8 = 7 \cdot x + x (- y)$

Rewrite using the commutative property of multiplication.

$3 y - 8 = 7 x - x y$

Solve the equation.

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

$3 y - 8 + x y = 7 x$

Add $8$ to both sides of the equation.

$3 y + x y = 7 x + 8$

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

Factor $y$ out of $3 y$ .

$y \cdot 3 + x y = 7 x + 8$

Factor $y$ out of $x y$ .

$y \cdot 3 + y x = 7 x + 8$

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

$y (3 + x) = 7 x + 8$

Divide each term in $y (3 + x) = 7 x + 8$ by $3 + x$ and simplify.

Divide each term in $y (3 + x) = 7 x + 8$ by $3 + x$ .

$\frac{y (3 + x)}{3 + x} = \frac{7 x}{3 + x} + \frac{8}{3 + x}$

Simplify the left side.

Cancel the common factor of $3 + x$ .

Cancel the common factor.

$\frac{y (3 + x)}{3 + x} = \frac{7 x}{3 + x} + \frac{8}{3 + x}$

Divide $y$ by $1$ .

$y = \frac{7 x}{3 + x} + \frac{8}{3 + x}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{7 x + 8}{3 + x}$

Step 3

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

$f^{- 1} (x) = \frac{7 x + 8}{3 + x}$

Step 4

Verify if $f^{- 1} (x) = \frac{7 x + 8}{3 + x}$ is the inverse of $f (x) = \frac{3 x - 8}{7 - x}$ .

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 $f^{- 1} (\frac{3 x - 8}{7 - x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{7 (\frac{3 x - 8}{7 - x}) + 8}{3 + \frac{3 x - 8}{7 - x}}$

Remove parentheses.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{7 (\frac{3 x - 8}{7 - x}) + 8}{3 + \frac{3 x - 8}{7 - x}}$

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

Multiply $\frac{7 \frac{3 x - 8}{7 - x} + 8}{3 + \frac{3 x - 8}{7 - x}}$ by $\frac{7 - x}{7 - x}$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{7 - x}{7 - x} \cdot \frac{7 (\frac{3 x - 8}{7 - x}) + 8}{3 + \frac{3 x - 8}{7 - x}}$

Combine.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (7 (\frac{3 x - 8}{7 - x}) + 8)}{(7 - x) (3 + \frac{3 x - 8}{7 - x})}$

Apply the distributive property.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (7 (\frac{3 x - 8}{7 - x})) + (7 - x) \cdot 8}{(7 - x) \cdot 3 + (7 - x) (\frac{3 x - 8}{7 - x})}$

Cancel the common factor of $7 - x$ .

Cancel the common factor.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (7 (\frac{3 x - 8}{7 - x})) + (7 - x) \cdot 8}{(7 - x) \cdot 3 + (7 - x) (\frac{3 x - 8}{7 - x})}$

Rewrite the expression.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (7 (\frac{3 x - 8}{7 - x})) + (7 - x) \cdot 8}{(7 - x) \cdot 3 + 3 x - 8}$

Simplify the numerator.

Factor $7 - x$ out of $(7 - x) \cdot 7 \frac{3 x - 8}{7 - x} + (7 - x) \cdot 8$ .

Factor $7 - x$ out of $(7 - x) \cdot 7 \frac{3 x - 8}{7 - x}$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (7 (\frac{3 x - 8}{7 - x})) + (7 - x) \cdot 8}{(7 - x) \cdot 3 + 3 x - 8}$

Factor $7 - x$ out of $(7 - x) \cdot 8$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (7 (\frac{3 x - 8}{7 - x})) + (7 - x) (8)}{(7 - x) \cdot 3 + 3 x - 8}$

Factor $7 - x$ out of $(7 - x) (7 \frac{3 x - 8}{7 - x}) + (7 - x) (8)$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (7 (\frac{3 x - 8}{7 - x}) + 8)}{(7 - x) \cdot 3 + 3 x - 8}$

Combine $7$ and $\frac{3 x - 8}{7 - x}$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{7 (3 x - 8)}{7 - x} + 8)}{(7 - x) \cdot 3 + 3 x - 8}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{7 - x}{7 - x}$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{7 (3 x - 8)}{7 - x} + \frac{8 (7 - x)}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{7 (3 x - 8) + 8 (7 - x)}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Rewrite $\frac{7 (3 x - 8) + 8 (7 - x)}{7 - x}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{7 (3 x) + 7 \cdot - 8 + 8 (7 - x)}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Multiply $3$ by $7$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{21 x + 7 \cdot - 8 + 8 (7 - x)}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Multiply $7$ by $- 8$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{21 x - 56 + 8 (7 - x)}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Apply the distributive property.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{21 x - 56 + 8 \cdot 7 + 8 (- x)}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Multiply $8$ by $7$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{21 x - 56 + 56 + 8 (- x)}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Multiply $- 1$ by $8$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{21 x - 56 + 56 - 8 x}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Subtract $8 x$ from $21 x$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{13 x - 56 + 56}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Add $- 56$ and $56$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{13 x + 0}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Add $13 x$ and $0$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{13 x}{7 - x})}{(7 - x) \cdot 3 + 3 x - 8}$

Simplify the denominator.

Apply the distributive property.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{13 x}{7 - x})}{7 \cdot 3 - x \cdot 3 + 3 x - 8}$

Multiply $7$ by $3$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{13 x}{7 - x})}{21 - x \cdot 3 + 3 x - 8}$

Multiply $3$ by $- 1$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{13 x}{7 - x})}{21 - 3 x + 3 x - 8}$

Subtract $8$ from $21$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{13 x}{7 - x})}{- 3 x + 3 x + 13}$

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

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{13 x}{7 - x})}{0 + 13}$

Add $0$ and $13$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{(7 - x) (\frac{13 x}{7 - x})}{13}$

Reduce the expression by cancelling the common factors.

Factor $\frac{13 x}{7 - x}$ out of $\frac{(7 - x) \frac{13 x}{7 - x}}{13}$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{13 x}{7 - x} \cdot \frac{7 - x}{13}$

Cancel the common factor of $13$ .

Factor $13$ out of $13 x$ .

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{13 (x)}{7 - x} \cdot \frac{7 - x}{13}$

Cancel the common factor.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{13 x}{7 - x} \cdot \frac{7 - x}{13}$

Rewrite the expression.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{x}{7 - x} \cdot (7 - x)$

Cancel the common factor of $7 - x$ .

Cancel the common factor.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = \frac{x}{7 - x} \cdot (7 - x)$

Rewrite the expression.

$f^{- 1} (\frac{3 x - 8}{7 - x}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{7 x + 8}{3 + x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{3 (\frac{7 x + 8}{3 + x}) - 8}{7 - (\frac{7 x + 8}{3 + x})}$

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

Multiply $\frac{3 \frac{7 x + 8}{3 + x} - 8}{7 - \frac{7 x + 8}{3 + x}}$ by $\frac{3 + x}{3 + x}$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{3 + x}{3 + x} \cdot \frac{3 (\frac{7 x + 8}{3 + x}) - 8}{7 - \frac{7 x + 8}{3 + x}}$

Combine.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (3 (\frac{7 x + 8}{3 + x}) - 8)}{(3 + x) (7 - \frac{7 x + 8}{3 + x})}$

Apply the distributive property.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (3 (\frac{7 x + 8}{3 + x})) + (3 + x) \cdot - 8}{(3 + x) \cdot 7 + (3 + x) (- \frac{7 x + 8}{3 + x})}$

Cancel the common factor of $3 + x$ .

Move the leading negative in $- \frac{7 x + 8}{3 + x}$ into the numerator.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (3 (\frac{7 x + 8}{3 + x})) + (3 + x) \cdot - 8}{(3 + x) \cdot 7 + (3 + x) (\frac{- (7 x + 8)}{3 + x})}$

Cancel the common factor.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (3 (\frac{7 x + 8}{3 + x})) + (3 + x) \cdot - 8}{(3 + x) \cdot 7 + (3 + x) (\frac{- (7 x + 8)}{3 + x})}$

Rewrite the expression.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (3 (\frac{7 x + 8}{3 + x})) + (3 + x) \cdot - 8}{(3 + x) \cdot 7 - (7 x + 8)}$

Simplify the numerator.

Factor $3 + x$ out of $(3 + x) \cdot 3 \frac{7 x + 8}{3 + x} + (3 + x) \cdot - 8$ .

Factor $3 + x$ out of $(3 + x) \cdot 3 \frac{7 x + 8}{3 + x}$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (3 (\frac{7 x + 8}{3 + x})) + (3 + x) \cdot - 8}{(3 + x) \cdot 7 - (7 x + 8)}$

Factor $3 + x$ out of $(3 + x) \cdot - 8$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (3 (\frac{7 x + 8}{3 + x})) + (3 + x) (- 8)}{(3 + x) \cdot 7 - (7 x + 8)}$

Factor $3 + x$ out of $(3 + x) (3 \frac{7 x + 8}{3 + x}) + (3 + x) (- 8)$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (3 (\frac{7 x + 8}{3 + x}) - 8)}{(3 + x) \cdot 7 - (7 x + 8)}$

Combine $3$ and $\frac{7 x + 8}{3 + x}$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{3 (7 x + 8)}{3 + x} - 8)}{(3 + x) \cdot 7 - (7 x + 8)}$

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3 + x}{3 + x}$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{3 (7 x + 8)}{3 + x} - 8 \cdot \frac{3 + x}{3 + x})}{(3 + x) \cdot 7 - (7 x + 8)}$

Combine $- 8$ and $\frac{3 + x}{3 + x}$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{3 (7 x + 8)}{3 + x} + \frac{- 8 (3 + x)}{3 + x})}{(3 + x) \cdot 7 - (7 x + 8)}$

Combine the numerators over the common denominator.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{3 (7 x + 8) - 8 (3 + x)}{3 + x})}{(3 + x) \cdot 7 - (7 x + 8)}$

Reorder terms.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{3 (7 x + 8) - 8 (3 + x)}{x + 3})}{(3 + x) \cdot 7 - (7 x + 8)}$

Rewrite $\frac{3 (7 x + 8) - 8 (3 + x)}{x + 3}$ in a factored form.

Apply the distributive property.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{3 (7 x) + 3 \cdot 8 - 8 (3 + x)}{x + 3})}{(3 + x) \cdot 7 - (7 x + 8)}$

Multiply $7$ by $3$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{21 x + 3 \cdot 8 - 8 (3 + x)}{x + 3})}{(3 + x) \cdot 7 - (7 x + 8)}$

Multiply $3$ by $8$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{21 x + 24 - 8 (3 + x)}{x + 3})}{(3 + x) \cdot 7 - (7 x + 8)}$

Apply the distributive property.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{21 x + 24 - 8 \cdot 3 - 8 x}{x + 3})}{(3 + x) \cdot 7 - (7 x + 8)}$

Multiply $- 8$ by $3$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{21 x + 24 - 24 - 8 x}{x + 3})}{(3 + x) \cdot 7 - (7 x + 8)}$

Subtract $8 x$ from $21 x$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x + 24 - 24}{x + 3})}{(3 + x) \cdot 7 - (7 x + 8)}$

Subtract $24$ from $24$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x + 0}{x + 3})}{(3 + x) \cdot 7 - (7 x + 8)}$

Add $13 x$ and $0$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x}{x + 3})}{(3 + x) \cdot 7 - (7 x + 8)}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x}{x + 3})}{3 \cdot 7 + x \cdot 7 - (7 x + 8)}$

Multiply $3$ by $7$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x}{x + 3})}{21 + x \cdot 7 - (7 x + 8)}$

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

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x}{x + 3})}{21 + 7 \cdot x - (7 x + 8)}$

Apply the distributive property.

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x}{x + 3})}{21 + 7 x - (7 x) - 1 \cdot 8}$

Multiply $7$ by $- 1$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x}{x + 3})}{21 + 7 x - 7 x - 1 \cdot 8}$

Multiply $- 1$ by $8$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x}{x + 3})}{21 + 7 x - 7 x - 8}$

Subtract $8$ from $21$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x}{x + 3})}{7 x - 7 x + 13}$

Subtract $7 x$ from $7 x$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x}{x + 3})}{0 + 13}$

Add $0$ and $13$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{(3 + x) (\frac{13 x}{x + 3})}{13}$

Simplify terms.

Factor $\frac{13 x}{x + 3}$ out of $\frac{(3 + x) \frac{13 x}{x + 3}}{13}$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{13 x}{x + 3} \cdot \frac{3 + x}{13}$

Cancel the common factor of $13$ .

Factor $13$ out of $13 x$ .

$f (\frac{7 x + 8}{3 + x}) = \frac{13 (x)}{x + 3} \cdot \frac{3 + x}{13}$

Cancel the common factor.

$f (\frac{7 x + 8}{3 + x}) = \frac{13 x}{x + 3} \cdot \frac{3 + x}{13}$

Rewrite the expression.

$f (\frac{7 x + 8}{3 + x}) = \frac{x}{x + 3} \cdot (3 + x)$

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

$f (\frac{7 x + 8}{3 + x}) = \frac{x (3 + x)}{x + 3}$

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

Reorder terms.

$f (\frac{7 x + 8}{3 + x}) = \frac{x (x + 3)}{x + 3}$

Cancel the common factor.

$f (\frac{7 x + 8}{3 + x}) = \frac{x (x + 3)}{x + 3}$

Divide $x$ by $1$ .

$f (\frac{7 x + 8}{3 + x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x + 8}{3 + x}$ is the inverse of $f (x) = \frac{3 x - 8}{7 - x}$ .

$f^{- 1} (x) = \frac{7 x + 8}{3 + x}$

Find the Inverse (3x-8)/(7-x)

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