Find the Inverse f(x)=-1/3x-2/3

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

Write $f (x) = - \frac{1}{3} x - \frac{2}{3}$ as an equation.

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

Interchange the variables.

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

Solve for $y$ .

Rewrite the equation as $- \frac{1}{3} y - \frac{2}{3} = x$ .

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

Combine $y$ and $\frac{1}{3}$ .

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

Add $\frac{2}{3}$ to both sides of the equation.

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

Multiply both sides of the equation by $- 3$ .

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

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

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

Multiply $y$ by $1$ .

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

Simplify the right side.

Simplify $- 3 (x + \frac{2}{3})$ .

Apply the distributive property.

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

Cancel the common factor of $3$ .

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

$y = - 3 x - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$y = - 3 x - 2$

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

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

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

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

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

Simplify each term.

Combine $x$ and $\frac{1}{3}$ .

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

Apply the distributive property.

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 2$ .

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

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

Subtract $2$ from $2$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Factor $3$ out of $- 3 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- \frac{1}{3} \cdot - 2$ .

Multiply $- 2$ by $- 1$ .

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

Combine $2$ and $\frac{1}{3}$ .

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

Combine the opposite terms in $x + \frac{2}{3} - \frac{2}{3}$ .

Combine the numerators over the common denominator.

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

Subtract $2$ from $2$ .

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

Divide $0$ by $3$ .

$f (- 3 x - 2) = x + 0$

Add $x$ and $0$ .

$f (- 3 x - 2) = x$

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

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

Find the Inverse f(x)=-1/3x-2/3

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