Find the Inverse -3/7x-6/7

$- \frac{3}{7} x - \frac{6}{7}$

Step 1

Interchange the variables.

$x = - \frac{3}{7} y - \frac{6}{7}$

Step 2

Solve for $y$ .

Rewrite the equation as $- \frac{3}{7} y - \frac{6}{7} = x$ .

$- \frac{3}{7} y - \frac{6}{7} = x$

Simplify each term.

Combine $y$ and $\frac{3}{7}$ .

$- \frac{y \cdot 3}{7} - \frac{6}{7} = x$

Move $3$ to the left of $y$ .

$- \frac{3 y}{7} - \frac{6}{7} = x$

Add $\frac{6}{7}$ to both sides of the equation.

$- \frac{3 y}{7} = x + \frac{6}{7}$

Multiply both sides of the equation by $- \frac{7}{3}$ .

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{7}{3} (- \frac{3 y}{7})$ .

Cancel the common factor of $7$ .

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

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

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

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

Factor $7$ out of $- 7$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 1}{3} (- 3 y) = - \frac{7}{3} (x + \frac{6}{7})$

Cancel the common factor of $3$ .

Factor $3$ out of $- 3 y$ .

$\frac{- 1}{3} (3 (- y)) = - \frac{7}{3} (x + \frac{6}{7})$

Cancel the common factor.

$\frac{- 1}{3} (3 (- y)) = - \frac{7}{3} (x + \frac{6}{7})$

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{7}{3} (x + \frac{6}{7})$

Multiply $y$ by $1$ .

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

Simplify the right side.

Simplify $- \frac{7}{3} (x + \frac{6}{7})$ .

Simplify terms.

Apply the distributive property.

$y = - \frac{7}{3} x - \frac{7}{3} \cdot \frac{6}{7}$

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

$y = - \frac{x \cdot 7}{3} - \frac{7}{3} \cdot \frac{6}{7}$

Cancel the common factor of $7$ .

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

$y = - \frac{x \cdot 7}{3} + \frac{- 7}{3} \cdot \frac{6}{7}$

Factor $7$ out of $- 7$ .

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

Cancel the common factor.

$y = - \frac{x \cdot 7}{3} + \frac{7 \cdot - 1}{3} \cdot \frac{6}{7}$

Rewrite the expression.

$y = - \frac{x \cdot 7}{3} + \frac{- 1}{3} \cdot 6$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $2$ .

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

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

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

Step 3

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

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

Step 4

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

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 $f^{- 1} (- \frac{3}{7} x - \frac{6}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Simplify each term.

Simplify the numerator.

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

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

Move $3$ to the left of $x$ .

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

Factor $- 1$ out of $- \frac{3 x}{7} - \frac{6}{7}$ .

Factor $- 1$ out of $- \frac{3 x}{7}$ .

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

Factor $- 1$ out of $- \frac{6}{7}$ .

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

Factor $- 1$ out of $- (\frac{3 x}{7}) - (\frac{6}{7})$ .

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

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

Combine exponents.

Factor out negative.

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

Multiply $7$ by $- 1$ .

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

Combine the numerators over the common denominator.

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

Factor $3$ out of $3 x + 6$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $6$ .

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

Factor $3$ out of $3 x + 3 \cdot 2$ .

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

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

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

Multiply $- 7$ by $3$ .

$f^{- 1} (- \frac{3}{7} x - \frac{6}{7}) = - \frac{\frac{- 21 (x + 2)}{7}}{3} - 2$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 21 (x + 2)}{7}$ by cancelling the common factors.

Factor $7$ out of $- 21 (x + 2)$ .

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

Factor $7$ out of $7$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 3 (x + 2)$ by $1$ .

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

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

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

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- (x + 2)$ by $1$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Apply the distributive property.

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

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 1$ by $- 2$ .

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

Combine the opposite terms in $x + 2 - 2$ .

Subtract $2$ from $2$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $3$ .

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

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

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $7$ .

Factor $7$ out of $- 7 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- \frac{3}{7} \cdot - 2$ .

Multiply $- 2$ by $- 1$ .

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

Combine $2$ and $\frac{3}{7}$ .

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

Multiply $2$ by $3$ .

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

Combine the opposite terms in $x + \frac{6}{7} - \frac{6}{7}$ .

Combine the numerators over the common denominator.

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

Subtract $6$ from $6$ .

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

Divide $0$ by $7$ .

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

Add $x$ and $0$ .

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

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

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

Find the Inverse -3/7x-6/7

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