Find the Inverse -2/3x-5

$- \frac{2}{3} x - 5$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

Rewrite the equation as $- \frac{2}{3} y - 5 = x$ .

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

Simplify each term.

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

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

Move $2$ to the left of $y$ .

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

Add $5$ to both sides of the equation.

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

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

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

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $3$ .

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

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

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Factor $2$ out of $- 2 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

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

Multiply $y$ by $1$ .

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

Simplify the right side.

Simplify $- \frac{3}{2} (x + 5)$ .

Apply the distributive property.

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

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

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

Multiply $- \frac{3}{2} \cdot 5$ .

Multiply $5$ by $- 1$ .

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

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

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

Multiply $- 5$ by $3$ .

$y = - \frac{x \cdot 3}{2} + \frac{- 15}{2}$

Simplify each term.

Move $3$ to the left of $x$ .

$y = - \frac{3 \cdot x}{2} + \frac{- 15}{2}$

Move the negative in front of the fraction.

$y = - \frac{3 x}{2} - \frac{15}{2}$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = - \frac{3 x}{2} - \frac{15}{2}$ is the inverse of $f (x) = - \frac{2}{3} x - 5$ .

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

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

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

Move $2$ to the left of $x$ .

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

Apply the distributive property.

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 2$ by $- 1$ .

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

Multiply $- 3$ by $- 5$ .

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

Simplify terms.

Combine the opposite terms in $2 x + 15 - 15$ .

Subtract $15$ from $15$ .

$f^{- 1} (- \frac{2}{3} x - 5) = \frac{2 x + 0}{2}$

Add $2 x$ and $0$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

Evaluate $f (- \frac{3 x}{2} - \frac{15}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $2$ .

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

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

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

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

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Cancel the common factor of $2$ .

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

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

Move the leading negative in $- \frac{15}{2}$ into the numerator.

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

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Factor $3$ out of $- 15$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 5$ .

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

Combine the opposite terms in $x + 5 - 5$ .

Subtract $5$ from $5$ .

$f (- \frac{3 x}{2} - \frac{15}{2}) = x + 0$

Add $x$ and $0$ .

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

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{3 x}{2} - \frac{15}{2}$ is the inverse of $f (x) = - \frac{2}{3} x - 5$ .

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

Find the Inverse -2/3x-5

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