Find the Inverse (3x-6)/5

$\frac{(3 x - 6)}{5}$

Step 1

Interchange the variables.

$x = \frac{3 y - 6}{5}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{3 y - 6}{5} = x$ .

$\frac{3 y - 6}{5} = x$

Multiply both sides by $5$ .

$\frac{3 y - 6}{5} \cdot 5 = x \cdot 5$

Simplify.

Simplify the left side.

Simplify $\frac{3 y - 6}{5} \cdot 5$ .

Factor $3$ out of $3 y - 6$ .

Factor $3$ out of $3 y$ .

$\frac{3 (y) - 6}{5} \cdot 5 = x \cdot 5$

Factor $3$ out of $- 6$ .

$\frac{3 y + 3 \cdot - 2}{5} \cdot 5 = x \cdot 5$

Factor $3$ out of $3 y + 3 \cdot - 2$ .

$\frac{3 (y - 2)}{5} \cdot 5 = x \cdot 5$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{3 (y - 2)}{5} \cdot 5 = x \cdot 5$

Rewrite the expression.

$3 (y - 2) = x \cdot 5$

Apply the distributive property.

$3 y + 3 \cdot - 2 = x \cdot 5$

Multiply $3$ by $- 2$ .

$3 y - 6 = x \cdot 5$

Simplify the right side.

Move $5$ to the left of $x$ .

$3 y - 6 = 5 x$

Solve for $y$ .

Add $6$ to both sides of the equation.

$3 y = 5 x + 6$

Divide each term in $3 y = 5 x + 6$ by $3$ and simplify.

Divide each term in $3 y = 5 x + 6$ by $3$ .

$\frac{3 y}{3} = \frac{5 x}{3} + \frac{6}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{5 x}{3} + \frac{6}{3}$

Divide $y$ by $1$ .

$y = \frac{5 x}{3} + \frac{6}{3}$

Simplify the right side.

Divide $6$ by $3$ .

$y = \frac{5 x}{3} + 2$

Step 3

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

$f^{- 1} (x) = \frac{5 x}{3} + 2$

Step 4

Verify if $f^{- 1} (x) = \frac{5 x}{3} + 2$ is the inverse of $f (x) = \frac{3 x - 6}{5}$ .

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 $f^{- 1} (\frac{3 x - 6}{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{5 (\frac{3 x - 6}{5})}{3} + 2$

Simplify each term.

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

Factor $3$ out of $3 x$ .

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{5 (\frac{3 (x) - 6}{5})}{3} + 2$

Factor $3$ out of $- 6$ .

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{5 (\frac{3 x + 3 \cdot - 2}{5})}{3} + 2$

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

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{5 (\frac{3 (x - 2)}{5})}{3} + 2$

Combine $5$ and $\frac{3 (x - 2)}{5}$ .

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{\frac{5 (3 (x - 2))}{5}}{3} + 2$

Multiply $5$ by $3$ .

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{\frac{15 (x - 2)}{5}}{3} + 2$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{15 (x - 2)}{5}$ by cancelling the common factors.

Factor $5$ out of $15 (x - 2)$ .

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{\frac{5 (3 (x - 2))}{5}}{3} + 2$

Factor $5$ out of $5$ .

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{\frac{5 (3 (x - 2))}{5 (1)}}{3} + 2$

Cancel the common factor.

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{\frac{5 (3 (x - 2))}{5 \cdot 1}}{3} + 2$

Rewrite the expression.

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{\frac{3 (x - 2)}{1}}{3} + 2$

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

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{3 (x - 2)}{3} + 2$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\frac{3 x - 6}{5}) = \frac{3 (x - 2)}{3} + 2$

Divide $x - 2$ by $1$ .

$f^{- 1} (\frac{3 x - 6}{5}) = x - 2 + 2$

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

Add $- 2$ and $2$ .

$f^{- 1} (\frac{3 x - 6}{5}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{3 x - 6}{5}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{5 x}{3} + 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x}{3} + 2) = \frac{3 (\frac{5 x}{3} + 2) - 6}{5}$

Simplify the numerator.

Factor $3$ out of $3 (\frac{5 x}{3} + 2) - 6$ .

Factor $3$ out of $- 6$ .

$f (\frac{5 x}{3} + 2) = \frac{3 (\frac{5 x}{3} + 2) + 3 \cdot - 2}{5}$

Factor $3$ out of $3 (\frac{5 x}{3} + 2) + 3 \cdot - 2$ .

$f (\frac{5 x}{3} + 2) = \frac{3 (\frac{5 x}{3} + 2 - 2)}{5}$

Subtract $2$ from $2$ .

$f (\frac{5 x}{3} + 2) = \frac{3 (\frac{5 x}{3} + 0)}{5}$

Add $\frac{5 x}{3}$ and $0$ .

$f (\frac{5 x}{3} + 2) = \frac{3 (\frac{5 x}{3})}{5}$

Combine $3$ and $\frac{5 x}{3}$ .

$f (\frac{5 x}{3} + 2) = \frac{\frac{3 (5 x)}{3}}{5}$

Multiply $3$ by $5$ .

$f (\frac{5 x}{3} + 2) = \frac{\frac{15 x}{3}}{5}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{15 x}{3}$ by cancelling the common factors.

Factor $3$ out of $15 x$ .

$f (\frac{5 x}{3} + 2) = \frac{\frac{3 (5 x)}{3}}{5}$

Factor $3$ out of $3$ .

$f (\frac{5 x}{3} + 2) = \frac{\frac{3 (5 x)}{3 (1)}}{5}$

Cancel the common factor.

$f (\frac{5 x}{3} + 2) = \frac{\frac{3 (5 x)}{3 \cdot 1}}{5}$

Rewrite the expression.

$f (\frac{5 x}{3} + 2) = \frac{\frac{5 x}{1}}{5}$

Divide $5 x$ by $1$ .

$f (\frac{5 x}{3} + 2) = \frac{5 x}{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\frac{5 x}{3} + 2) = \frac{5 x}{5}$

Divide $x$ by $1$ .

$f (\frac{5 x}{3} + 2) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x}{3} + 2$ is the inverse of $f (x) = \frac{3 x - 6}{5}$ .

$f^{- 1} (x) = \frac{5 x}{3} + 2$

Find the Inverse (3x-6)/5

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