Find the Inverse (6x-7)/(2x-2)

$\frac{6 x - 7}{2 x - 2}$

Step 1

Interchange the variables.

$x = \frac{6 y - 7}{2 y - 2}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{6 y - 7}{2 y - 2} = x$ .

$\frac{6 y - 7}{2 y - 2} = x$

Factor $2$ out of $2 y - 2$ .

Factor $2$ out of $2 y$ .

$\frac{6 y - 7}{2 (y) - 2} = x$

Factor $2$ out of $- 2$ .

$\frac{6 y - 7}{2 (y) + 2 (- 1)} = x$

Factor $2$ out of $2 (y) + 2 (- 1)$ .

$\frac{6 y - 7}{2 (y - 1)} = x$

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.

$2 (y - 1), 1$

The LCM of one and any expression is the expression.

$2 (y - 1)$

Multiply each term in $\frac{6 y - 7}{2 (y - 1)} = x$ by $2 (y - 1)$ to eliminate the fractions.

Multiply each term in $\frac{6 y - 7}{2 (y - 1)} = x$ by $2 (y - 1)$ .

$\frac{6 y - 7}{2 (y - 1)} (2 (y - 1)) = x (2 (y - 1))$

Simplify the left side.

Rewrite using the commutative property of multiplication.

$2 \frac{6 y - 7}{2 (y - 1)} (y - 1) = x (2 (y - 1))$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{6 y - 7}{2 (y - 1)} (y - 1) = x (2 (y - 1))$

Rewrite the expression.

$\frac{6 y - 7}{y - 1} (y - 1) = x (2 (y - 1))$

Cancel the common factor of $y - 1$ .

Cancel the common factor.

$\frac{6 y - 7}{y - 1} (y - 1) = x (2 (y - 1))$

Rewrite the expression.

$6 y - 7 = x (2 (y - 1))$

Simplify the right side.

Rewrite using the commutative property of multiplication.

$6 y - 7 = 2 x (y - 1)$

Apply the distributive property.

$6 y - 7 = 2 x y + 2 x \cdot - 1$

Multiply $- 1$ by $2$ .

$6 y - 7 = 2 x y - 2 x$

Solve the equation.

Subtract $2 x y$ from both sides of the equation.

$6 y - 7 - 2 x y = - 2 x$

Add $7$ to both sides of the equation.

$6 y - 2 x y = - 2 x + 7$

Factor $2 y$ out of $6 y - 2 x y$ .

Factor $2 y$ out of $6 y$ .

$2 y (3) - 2 x y = - 2 x + 7$

Factor $2 y$ out of $- 2 x y$ .

$2 y (3) + 2 y (- x) = - 2 x + 7$

Factor $2 y$ out of $2 y (3) + 2 y (- x)$ .

$2 y (3 - x) = - 2 x + 7$

Divide each term in $2 y (3 - x) = - 2 x + 7$ by $2 (3 - x)$ and simplify.

Divide each term in $2 y (3 - x) = - 2 x + 7$ by $2 (3 - x)$ .

$\frac{2 y (3 - x)}{2 (3 - x)} = \frac{- 2 x}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y (3 - x)}{2 (3 - x)} = \frac{- 2 x}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Rewrite the expression.

$\frac{y (3 - x)}{3 - x} = \frac{- 2 x}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Cancel the common factor of $3 - x$ .

Cancel the common factor.

$\frac{y (3 - x)}{3 - x} = \frac{- 2 x}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Divide $y$ by $1$ .

$y = \frac{- 2 x}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 2$ and $2$ .

Factor $2$ out of $- 2 x$ .

$y = \frac{2 (- x)}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Cancel the common factors.

Cancel the common factor.

$y = \frac{2 (- x)}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Rewrite the expression.

$y = \frac{- x}{3 - x} + \frac{7}{2 (3 - x)}$

Move the negative in front of the fraction.

$y = - \frac{x}{3 - x} + \frac{7}{2 (3 - x)}$

To write $- \frac{x}{3 - x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{x}{3 - x} \cdot \frac{2}{2} + \frac{7}{2 (3 - x)}$

Write each expression with a common denominator of $(3 - x) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x}{3 - x}$ by $\frac{2}{2}$ .

$y = - \frac{x \cdot 2}{(3 - x) \cdot 2} + \frac{7}{2 (3 - x)}$

Reorder the factors of $(3 - x) \cdot 2$ .

$y = - \frac{x \cdot 2}{2 (3 - x)} + \frac{7}{2 (3 - x)}$

Combine the numerators over the common denominator.

$y = \frac{- x \cdot 2 + 7}{2 (3 - x)}$

Multiply $2$ by $- 1$ .

$y = \frac{- 2 x + 7}{2 (3 - x)}$

Factor $- 1$ out of $- 2 x$ .

$y = \frac{- (2 x) + 7}{2 (3 - x)}$

Rewrite $7$ as $- 1 (- 7)$ .

$y = \frac{- (2 x) - 1 (- 7)}{2 (3 - x)}$

Factor $- 1$ out of $- (2 x) - 1 (- 7)$ .

$y = \frac{- (2 x - 7)}{2 (3 - x)}$

Simplify the expression.

Rewrite $- (2 x - 7)$ as $- 1 (2 x - 7)$ .

$y = \frac{- 1 (2 x - 7)}{2 (3 - x)}$

Move the negative in front of the fraction.

$y = - \frac{2 x - 7}{2 (3 - x)}$

Step 3

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

$f^{- 1} (x) = - \frac{2 x - 7}{2 (3 - x)}$

Step 4

Verify if $f^{- 1} (x) = - \frac{2 x - 7}{2 (3 - x)}$ is the inverse of $f (x) = \frac{6 x - 7}{2 x - 2}$ .

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 $f^{- 1} (\frac{6 x - 7}{2 x - 2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 x - 2}) - 7}{2 (3 - (\frac{6 x - 7}{2 x - 2}))}$

Simplify with factoring out.

Factor $2$ out of $2 x - 2$ .

Factor $2$ out of $2 x$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x) - 2}) - 7}{2 (3 - \frac{6 x - 7}{2 x - 2})}$

Factor $2$ out of $- 2$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x) + 2 (- 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 x - 2})}$

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x - 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 x - 2})}$

Factor $2$ out of $2 x - 2$ .

Factor $2$ out of $2 x$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x - 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 (x) - 2})}$

Factor $2$ out of $- 2$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x - 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 (x) + 2 (- 1)})}$

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x - 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Simplify the numerator.

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{2 (\frac{6 x - 7}{2 (x - 1)}) - 7}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Rewrite the expression.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7}{x - 1} - 7}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7}{x - 1} - 7 \cdot \frac{x - 1}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Combine $- 7$ and $\frac{x - 1}{x - 1}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7}{x - 1} + \frac{- 7 (x - 1)}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7 - 7 (x - 1)}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Rewrite $\frac{6 x - 7 - 7 (x - 1)}{x - 1}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7 - 7 x - 7 \cdot - 1}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Multiply $- 7$ by $- 1$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{6 x - 7 - 7 x + 7}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Subtract $7 x$ from $6 x$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{- x - 7 + 7}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Add $- 7$ and $7$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{- x + 0}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Add $- x$ and $0$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{\frac{- x}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (3 - \frac{6 x - 7}{2 (x - 1)})}$

Simplify the denominator.

To write $3$ as a fraction with a common denominator, multiply by $\frac{2 (x - 1)}{2 (x - 1)}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (3 \cdot \frac{2 (x - 1)}{2 (x - 1)} - \frac{6 x - 7}{2 (x - 1)})}$

Combine $3$ and $\frac{2 (x - 1)}{2 (x - 1)}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{3 (2 (x - 1))}{2 (x - 1)} - \frac{6 x - 7}{2 (x - 1)})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{3 (2 (x - 1)) - (6 x - 7)}{2 (x - 1)})}$

Rewrite $\frac{3 (2 (x - 1)) - (6 x - 7)}{2 (x - 1)}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{3 (2 x + 2 \cdot - 1) - (6 x - 7)}{2 (x - 1)})}$

Multiply $2$ by $- 1$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{3 (2 x - 2) - (6 x - 7)}{2 (x - 1)})}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{3 (2 x) + 3 \cdot - 2 - (6 x - 7)}{2 (x - 1)})}$

Multiply $2$ by $3$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{6 x + 3 \cdot - 2 - (6 x - 7)}{2 (x - 1)})}$

Multiply $3$ by $- 2$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{6 x - 6 - (6 x - 7)}{2 (x - 1)})}$

Apply the distributive property.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{6 x - 6 - (6 x) + 7}{2 (x - 1)})}$

Multiply $6$ by $- 1$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{6 x - 6 - 6 x + 7}{2 (x - 1)})}$

Multiply $- 1$ by $- 7$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{6 x - 6 - 6 x + 7}{2 (x - 1)})}$

Subtract $6 x$ from $6 x$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{0 - 6 + 7}{2 (x - 1)})}$

Subtract $6$ from $0$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{- 6 + 7}{2 (x - 1)})}$

Add $- 6$ and $7$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{2 (\frac{1}{2 (x - 1)})}$

Simplify terms.

Combine $2$ and $\frac{1}{2 (x - 1)}$ .

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{\frac{2}{2 (x - 1)}}$

Reduce the expression $\frac{2}{2 (x - 1)}$ by cancelling the common factors.

Cancel the common factor.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{\frac{2}{2 (x - 1)}}$

Rewrite the expression.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - \frac{- \frac{x}{x - 1}}{\frac{1}{x - 1}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - (- \frac{x}{x - 1} \cdot (x - 1))$

Cancel the common factor of $x - 1$ .

Move the leading negative in $- \frac{x}{x - 1}$ into the numerator.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - (\frac{- x}{x - 1} \cdot (x - 1))$

Cancel the common factor.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = - (\frac{- x}{x - 1} \cdot (x - 1))$

Rewrite the expression.

$f^{- 1} (\frac{6 x - 7}{2 x - 2}) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{2 x - 7}{2 (3 - x)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{6 (- \frac{2 x - 7}{2 (3 - x)}) - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Simplify the numerator.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2 x - 7}{2 (3 - x)}$ into the numerator.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{6 (\frac{- (2 x - 7)}{2 (3 - x)}) - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Factor $2$ out of $6$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{2 (3) (\frac{- (2 x - 7)}{2 (3 - x)}) - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Cancel the common factor.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{2 \cdot (3 (\frac{- (2 x - 7)}{2 (3 - x)})) - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Rewrite the expression.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{3 (\frac{- (2 x - 7)}{3 - x}) - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Combine $3$ and $\frac{- (2 x - 7)}{3 - x}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{3 (- (2 x - 7))}{3 - x} - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Multiply $- 1$ by $3$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- 3 (2 x - 7)}{3 - x} - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Move the negative in front of the fraction.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{- \frac{(3) (2 x - 7)}{3 - x} - 7}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3 - x}{3 - x}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{- \frac{3 (2 x - 7)}{3 - x} - 7 \cdot \frac{3 - x}{3 - x}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

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

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{- \frac{3 (2 x - 7)}{3 - x} + \frac{- 7 (3 - x)}{3 - x}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Combine the numerators over the common denominator.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- 3 (2 x - 7) - 7 (3 - x)}{3 - x}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Reorder terms.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- 3 (2 x - 7) - 7 (3 - x)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Rewrite $\frac{- 3 (2 x - 7) - 7 (3 - x)}{- x + 3}$ in a factored form.

Factor $- 1$ out of $- 3 (2 x - 7) - 7 (3 - x)$ .

Reorder $3$ and $- x$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- 3 (2 x - 7) - 7 (- x + 3)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Factor $- 1$ out of $- 3 (2 x - 7)$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (3 (2 x - 7)) - 7 (- x + 3)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Factor $- 1$ out of $- 7 (- x + 3)$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (3 (2 x - 7)) - (7 (- x + 3))}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Factor $- 1$ out of $- (3 (2 x - 7)) - (7 (- x + 3))$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (3 (2 x - 7) + 7 (- x + 3))}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Apply the distributive property.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (3 (2 x) + 3 \cdot - 7 + 7 (- x + 3))}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Multiply $2$ by $3$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (6 x + 3 \cdot - 7 + 7 (- x + 3))}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Multiply $3$ by $- 7$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (6 x - 21 + 7 (- x + 3))}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Apply the distributive property.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (6 x - 21 + 7 (- x) + 7 \cdot 3)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Multiply $- 1$ by $7$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (6 x - 21 - 7 x + 7 \cdot 3)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Multiply $7$ by $3$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (6 x - 21 - 7 x + 21)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Subtract $7 x$ from $6 x$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (- x - 21 + 21)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Add $- 21$ and $21$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{- (- x + 0)}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Add $- x$ and $0$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{1 x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Combine exponents.

Factor out negative.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Multiply $- 1$ by $- 1$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{1 x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Multiply $x$ by $1$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) - 2}$

Simplify the denominator.

Factor $2$ out of $2 (- \frac{2 x - 7}{2 (3 - x)}) - 2$ .

Factor $2$ out of $- 2$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)}) + 2 (- 1)}$

Factor $2$ out of $2 (- \frac{2 x - 7}{2 (3 - x)}) + 2 (- 1)$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)} - 1)}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2 (3 - x)}{2 (3 - x)}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)} - 1 \cdot \frac{2 (3 - x)}{2 (3 - x)})}$

Combine $- 1$ and $\frac{2 (3 - x)}{2 (3 - x)}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (- \frac{2 x - 7}{2 (3 - x)} + \frac{- (2 (3 - x))}{2 (3 - x)})}$

Combine the numerators over the common denominator.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7) - (2 (3 - x))}{2 (3 - x)})}$

Rewrite $\frac{- (2 x - 7) - (2 (3 - x))}{2 (3 - x)}$ in a factored form.

Factor $- 1$ out of $- (2 x - 7) - 1 \cdot 2 (3 - x)$ .

Reorder $3$ and $- x$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7) - 1 \cdot (2 (- x + 3))}{2 (3 - x)})}$

Factor $- 1$ out of $- 1 \cdot 2 (- x + 3)$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7) - (2 (- x + 3))}{2 (3 - x)})}$

Factor $- 1$ out of $- (2 x - 7) - (2 (- x + 3))$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7 + 2 (- x + 3))}{2 (3 - x)})}$

Apply the distributive property.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7 + 2 (- x) + 2 \cdot 3)}{2 (3 - x)})}$

Multiply $- 1$ by $2$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7 - 2 x + 2 \cdot 3)}{2 (3 - x)})}$

Multiply $2$ by $3$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (2 x - 7 - 2 x + 6)}{2 (3 - x)})}$

Subtract $2 x$ from $2 x$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (0 - 7 + 6)}{2 (3 - x)})}$

Subtract $7$ from $0$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{- (- 7 + 6)}{2 (3 - x)})}$

Add $- 7$ and $6$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{1}{2 (3 - x)})}$

Multiply $- 1$ by $- 1$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{2 (\frac{1}{2 (3 - x)})}$

Simplify terms.

Combine $2$ and $\frac{1}{2 (3 - x)}$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{\frac{2}{2 (3 - x)}}$

Reduce the expression $\frac{2}{2 (3 - x)}$ by cancelling the common factors.

Cancel the common factor.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{\frac{2}{2 (3 - x)}}$

Rewrite the expression.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{\frac{x}{- x + 3}}{\frac{1}{3 - x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{x}{- x + 3} \cdot (3 - x)$

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

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{x (3 - x)}{- x + 3}$

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

Reorder terms.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{x (- x + 3)}{- x + 3}$

Cancel the common factor.

$f (- \frac{2 x - 7}{2 (3 - x)}) = \frac{x (- x + 3)}{- x + 3}$

Divide $x$ by $1$ .

$f (- \frac{2 x - 7}{2 (3 - x)}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{2 x - 7}{2 (3 - x)}$ is the inverse of $f (x) = \frac{6 x - 7}{2 x - 2}$ .

$f^{- 1} (x) = - \frac{2 x - 7}{2 (3 - x)}$

Find the Inverse (6x-7)/(2x-2)

Related Questions

Support LCMGCF.com by disabling your adblocker.