Find the Inverse 2/3x-6

$\frac{2}{3} x - 6$

Step 1

Interchange the variables.

$x = \frac{2}{3} y - 6$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{2}{3} y - 6 = x$ .

$\frac{2}{3} y - 6 = x$

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

$\frac{2 y}{3} - 6 = x$

Add $6$ to both sides of the equation.

$\frac{2 y}{3} = x + 6$

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 y}{3} = \frac{3}{2} (x + 6)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{3}{2} \cdot \frac{2 y}{3}$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 y}{3} = \frac{3}{2} (x + 6)$

Rewrite the expression.

$\frac{1}{2} (2 y) = \frac{3}{2} (x + 6)$

Cancel the common factor of $2$ .

Factor $2$ out of $2 y$ .

$\frac{1}{2} (2 (y)) = \frac{3}{2} (x + 6)$

Cancel the common factor.

$\frac{1}{2} (2 y) = \frac{3}{2} (x + 6)$

Rewrite the expression.

$y = \frac{3}{2} (x + 6)$

Simplify the right side.

Simplify $\frac{3}{2} (x + 6)$ .

Apply the distributive property.

$y = \frac{3}{2} x + \frac{3}{2} \cdot 6$

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

$y = \frac{3 x}{2} + \frac{3}{2} \cdot 6$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $3$ .

$y = \frac{3 x}{2} + 9$

Step 3

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

$f^{- 1} (x) = \frac{3 x}{2} + 9$

Step 4

Verify if $f^{- 1} (x) = \frac{3 x}{2} + 9$ is the inverse of $f (x) = \frac{2}{3} x - 6$ .

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 $f^{- 1} (\frac{2}{3} x - 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2}{3} x - 6) = \frac{3 (\frac{2}{3} x - 6)}{2} + 9$

Simplify each term.

Simplify the numerator.

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

$f^{- 1} (\frac{2}{3} x - 6) = \frac{3 (\frac{2 x}{3} - 6)}{2} + 9$

Factor $2$ out of $\frac{2 x}{3} - 6$ .

Factor $2$ out of $\frac{2 x}{3}$ .

$f^{- 1} (\frac{2}{3} x - 6) = \frac{3 (2 (\frac{x}{3}) - 6)}{2} + 9$

Factor $2$ out of $- 6$ .

$f^{- 1} (\frac{2}{3} x - 6) = \frac{3 (2 (\frac{x}{3}) + 2 \cdot - 3)}{2} + 9$

Factor $2$ out of $2 \frac{x}{3} + 2 \cdot - 3$ .

$f^{- 1} (\frac{2}{3} x - 6) = \frac{3 (2 (\frac{x}{3} - 3))}{2} + 9$

$f^{- 1} (\frac{2}{3} x - 6) = \frac{3 \cdot (2 (\frac{x}{3} - 3))}{2} + 9$

Multiply $3$ by $2$ .

$f^{- 1} (\frac{2}{3} x - 6) = \frac{6 (\frac{x}{3} - 3)}{2} + 9$

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

$f^{- 1} (\frac{2}{3} x - 6) = \frac{6 (\frac{x}{3} - 3 \cdot \frac{3}{3})}{2} + 9$

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

$f^{- 1} (\frac{2}{3} x - 6) = \frac{6 (\frac{x}{3} + \frac{- 3 \cdot 3}{3})}{2} + 9$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{2}{3} x - 6) = \frac{6 (\frac{x - 3 \cdot 3}{3})}{2} + 9$

Multiply $- 3$ by $3$ .

$f^{- 1} (\frac{2}{3} x - 6) = \frac{6 (\frac{x - 9}{3})}{2} + 9$

Combine $6$ and $\frac{x - 9}{3}$ .

$f^{- 1} (\frac{2}{3} x - 6) = \frac{\frac{6 (x - 9)}{3}}{2} + 9$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{6 (x - 9)}{3}$ by cancelling the common factors.

Factor $3$ out of $6 (x - 9)$ .

$f^{- 1} (\frac{2}{3} x - 6) = \frac{\frac{3 (2 (x - 9))}{3}}{2} + 9$

Factor $3$ out of $3$ .

$f^{- 1} (\frac{2}{3} x - 6) = \frac{\frac{3 (2 (x - 9))}{3 (1)}}{2} + 9$

Cancel the common factor.

$f^{- 1} (\frac{2}{3} x - 6) = \frac{\frac{3 (2 (x - 9))}{3 \cdot 1}}{2} + 9$

Rewrite the expression.

$f^{- 1} (\frac{2}{3} x - 6) = \frac{\frac{2 (x - 9)}{1}}{2} + 9$

Divide $2 (x - 9)$ by $1$ .

$f^{- 1} (\frac{2}{3} x - 6) = \frac{2 (x - 9)}{2} + 9$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\frac{2}{3} x - 6) = \frac{2 (x - 9)}{2} + 9$

Divide $x - 9$ by $1$ .

$f^{- 1} (\frac{2}{3} x - 6) = x - 9 + 9$

Combine the opposite terms in $x - 9 + 9$ .

Add $- 9$ and $9$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (\frac{3 x}{2} + 9)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x}{2} + 9) = \frac{2}{3} \cdot (\frac{3 x}{2} + 9) - 6$

Simplify each term.

Apply the distributive property.

$f (\frac{3 x}{2} + 9) = \frac{2}{3} \cdot \frac{3 x}{2} + \frac{2}{3} \cdot 9 - 6$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{3 x}{2} + 9) = \frac{2}{3} \cdot \frac{3 x}{2} + \frac{2}{3} \cdot 9 - 6$

Rewrite the expression.

$f (\frac{3 x}{2} + 9) = \frac{1}{3} \cdot (3 x) + \frac{2}{3} \cdot 9 - 6$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$f (\frac{3 x}{2} + 9) = \frac{1}{3} \cdot (3 (x)) + \frac{2}{3} \cdot 9 - 6$

Cancel the common factor.

$f (\frac{3 x}{2} + 9) = \frac{1}{3} \cdot (3 x) + \frac{2}{3} \cdot 9 - 6$

Rewrite the expression.

$f (\frac{3 x}{2} + 9) = x + \frac{2}{3} \cdot 9 - 6$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$f (\frac{3 x}{2} + 9) = x + \frac{2}{3} \cdot (3 (3)) - 6$

Cancel the common factor.

$f (\frac{3 x}{2} + 9) = x + \frac{2}{3} \cdot (3 \cdot 3) - 6$

Rewrite the expression.

$f (\frac{3 x}{2} + 9) = x + 2 \cdot 3 - 6$

Multiply $2$ by $3$ .

$f (\frac{3 x}{2} + 9) = x + 6 - 6$

Combine the opposite terms in $x + 6 - 6$ .

Subtract $6$ from $6$ .

$f (\frac{3 x}{2} + 9) = x + 0$

Add $x$ and $0$ .

$f (\frac{3 x}{2} + 9) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{2} + 9$ is the inverse of $f (x) = \frac{2}{3} x - 6$ .

$f^{- 1} (x) = \frac{3 x}{2} + 9$

Find the Inverse 2/3x-6

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