Find the Inverse f(x)=-(6x)/7

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

Write $f (x) = - \frac{6 x}{7}$ as an equation.

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

Interchange the variables.

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

Solve for $y$ .

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

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

Multiply both sides of the equation by $- \frac{7}{6}$ .

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{7}{6} (- \frac{6 y}{7})$ .

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{7}{6}$ into the numerator.

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

Move the leading negative in $- \frac{6 y}{7}$ into the numerator.

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

Factor $7$ out of $- 7$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $6$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

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

Multiply $y$ by $1$ .

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

Simplify the right side.

Simplify $- \frac{7}{6} x$ .

Combine $x$ and $\frac{7}{6}$ .

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

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

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

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

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

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

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

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

Simplify the numerator.

Multiply $- 1$ by $7$ .

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

Combine $- 7$ and $\frac{6 x}{7}$ .

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

Multiply $- 7$ by $6$ .

$f^{- 1} (- \frac{6 x}{7}) = - \frac{\frac{- 42 x}{7}}{6}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 42 x}{7}$ by cancelling the common factors.

Factor $7$ out of $- 42 x$ .

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

Factor $7$ out of $7$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 6 x$ by $1$ .

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

Cancel the common factor of $- 6$ and $6$ .

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

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

Cancel the common factors.

Factor $6$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Simplify the numerator.

Multiply $- 1$ by $6$ .

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

Combine $- 6$ and $\frac{7 x}{6}$ .

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

Multiply $- 6$ by $7$ .

$f (- \frac{7 x}{6}) = - \frac{\frac{- 42 x}{6}}{7}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 42 x}{6}$ by cancelling the common factors.

Factor $6$ out of $- 42 x$ .

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

Factor $6$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 7 x$ by $1$ .

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

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

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

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

Cancel the common factors.

Factor $7$ out of $7$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- x$ by $1$ .

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

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

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

Find the Inverse f(x)=-(6x)/7

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