Find the Inverse h(x)=3/2(x-11)

$h (x) = \frac{3}{2} (x - 11)$

Step 1

Write $h (x) = \frac{3}{2} (x - 11)$ as an equation.

$y = \frac{3}{2} (x - 11)$

Step 2

Interchange the variables.

$x = \frac{3}{2} \cdot (y - 11)$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{3}{2} \cdot (y - 11) = x$ .

$\frac{3}{2} \cdot (y - 11) = x$

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

$\frac{2}{3} (\frac{3}{2} \cdot (y - 11)) = \frac{2}{3} x$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{2}{3} (\frac{3}{2} \cdot (y - 11))$ .

Apply the distributive property.

$\frac{2}{3} (\frac{3}{2} y + \frac{3}{2} \cdot - 11) = \frac{2}{3} x$

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

$\frac{2}{3} (\frac{3 y}{2} + \frac{3}{2} \cdot - 11) = \frac{2}{3} x$

Multiply $\frac{3}{2} \cdot - 11$ .

Combine $\frac{3}{2}$ and $- 11$ .

$\frac{2}{3} (\frac{3 y}{2} + \frac{3 \cdot - 11}{2}) = \frac{2}{3} x$

Multiply $3$ by $- 11$ .

$\frac{2}{3} (\frac{3 y}{2} + \frac{- 33}{2}) = \frac{2}{3} x$

Move the negative in front of the fraction.

$\frac{2}{3} (\frac{3 y}{2} - \frac{33}{2}) = \frac{2}{3} x$

Apply the distributive property.

$\frac{2}{3} \cdot \frac{3 y}{2} + \frac{2}{3} (- \frac{33}{2}) = \frac{2}{3} x$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 y}{2} + \frac{2}{3} (- \frac{33}{2}) = \frac{2}{3} x$

Rewrite the expression.

$\frac{1}{3} (3 y) + \frac{2}{3} (- \frac{33}{2}) = \frac{2}{3} x$

Cancel the common factor of $3$ .

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) + \frac{2}{3} (- \frac{33}{2}) = \frac{2}{3} x$

Cancel the common factor.

$\frac{1}{3} (3 y) + \frac{2}{3} (- \frac{33}{2}) = \frac{2}{3} x$

Rewrite the expression.

$y + \frac{2}{3} (- \frac{33}{2}) = \frac{2}{3} x$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{33}{2}$ into the numerator.

$y + \frac{2}{3} \cdot \frac{- 33}{2} = \frac{2}{3} x$

Cancel the common factor.

$y + \frac{2}{3} \cdot \frac{- 33}{2} = \frac{2}{3} x$

Rewrite the expression.

$y + \frac{1}{3} \cdot - 33 = \frac{2}{3} x$

Cancel the common factor of $3$ .

Factor $3$ out of $- 33$ .

$y + \frac{1}{3} \cdot (3 (- 11)) = \frac{2}{3} x$

Cancel the common factor.

$y + \frac{1}{3} \cdot (3 \cdot - 11) = \frac{2}{3} x$

Rewrite the expression.

$y - 11 = \frac{2}{3} x$

Simplify the right side.

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

$y - 11 = \frac{2 x}{3}$

Add $11$ to both sides of the equation.

$y = \frac{2 x}{3} + 11$

Step 4

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

$h^{- 1} (x) = \frac{2 x}{3} + 11$

Step 5

Verify if $h^{- 1} (x) = \frac{2 x}{3} + 11$ is the inverse of $h (x) = \frac{3}{2} \cdot (x - 11)$ .

To verify the inverse, check if $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ .

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

Set up the composite result function.

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

Evaluate $h^{- 1} (\frac{3}{2} \cdot (x - 11))$ by substituting in the value of $h$ into $h^{- 1}$ .

$h^{- 1} (\frac{3}{2} \cdot (x - 11)) = \frac{2 (\frac{3}{2} \cdot (x - 11))}{3} + 11$

Simplify each term.

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

$h^{- 1} (\frac{3}{2} \cdot (x - 11)) = \frac{\frac{2 \cdot 3}{2} \cdot (x - 11)}{3} + 11$

Multiply $2$ by $3$ .

$h^{- 1} (\frac{3}{2} \cdot (x - 11)) = \frac{\frac{6}{2} \cdot (x - 11)}{3} + 11$

Divide $6$ by $2$ .

$h^{- 1} (\frac{3}{2} \cdot (x - 11)) = \frac{3 (x - 11)}{3} + 11$

Cancel the common factor of $3$ .

Cancel the common factor.

$h^{- 1} (\frac{3}{2} \cdot (x - 11)) = \frac{3 (x - 11)}{3} + 11$

Divide $x - 11$ by $1$ .

$h^{- 1} (\frac{3}{2} \cdot (x - 11)) = x - 11 + 11$

Combine the opposite terms in $x - 11 + 11$ .

Add $- 11$ and $11$ .

$h^{- 1} (\frac{3}{2} \cdot (x - 11)) = x + 0$

Add $x$ and $0$ .

$h^{- 1} (\frac{3}{2} \cdot (x - 11)) = x$

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

Set up the composite result function.

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

Evaluate $h (\frac{2 x}{3} + 11)$ by substituting in the value of $h^{- 1}$ into $h$ .

$h (\frac{2 x}{3} + 11) = \frac{3}{2} \cdot ((\frac{2 x}{3} + 11) - 11)$

Combine the opposite terms in $(\frac{2 x}{3} + 11) - 11$ .

Subtract $11$ from $11$ .

$h (\frac{2 x}{3} + 11) = \frac{3}{2} \cdot (\frac{2 x}{3} + 0)$

Add $\frac{2 x}{3}$ and $0$ .

$h (\frac{2 x}{3} + 11) = \frac{3}{2} \cdot \frac{2 x}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$h (\frac{2 x}{3} + 11) = \frac{3}{2} \cdot \frac{2 x}{3}$

Rewrite the expression.

$h (\frac{2 x}{3} + 11) = \frac{1}{2} \cdot (2 x)$

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

$h (\frac{2 x}{3} + 11) = \frac{1}{2} \cdot (2 (x))$

Cancel the common factor.

$h (\frac{2 x}{3} + 11) = \frac{1}{2} \cdot (2 x)$

Rewrite the expression.

$h (\frac{2 x}{3} + 11) = x$

Since $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ , then $h^{- 1} (x) = \frac{2 x}{3} + 11$ is the inverse of $h (x) = \frac{3}{2} \cdot (x - 11)$ .

$h^{- 1} (x) = \frac{2 x}{3} + 11$

Find the Inverse h(x)=3/2(x-11)

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