Find the Inverse x-2/3

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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

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

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

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

Step 3

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

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

Step 4

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

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

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

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

Add $- \frac{2}{3}$ and $\frac{2}{3}$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

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

Combine the numerators over the common denominator.

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

Subtract $2$ from $2$ .

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

Divide $0$ by $3$ .

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

Add $x$ and $0$ .

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

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

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