Find the Inverse F(x)=-3/2x+7/2

$F (x) = - \frac{3}{2} x + \frac{7}{2}$

Step 1

Write $F (x) = - \frac{3}{2} x + \frac{7}{2}$ as an equation.

$y = - \frac{3}{2} x + \frac{7}{2}$

Step 2

Interchange the variables.

$x = - \frac{3}{2} y + \frac{7}{2}$

Step 3

Solve for $y$ .

Rewrite the equation as $- \frac{3}{2} y + \frac{7}{2} = x$ .

$- \frac{3}{2} y + \frac{7}{2} = x$

Simplify each term.

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

$- \frac{y \cdot 3}{2} + \frac{7}{2} = x$

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

$- \frac{3 y}{2} + \frac{7}{2} = x$

Subtract $\frac{7}{2}$ from both sides of the equation.

$- \frac{3 y}{2} = x - \frac{7}{2}$

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

$- \frac{2}{3} (- \frac{3 y}{2}) = - \frac{2}{3} (x - \frac{7}{2})$

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $2$ .

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

$\frac{- 2}{3} (- \frac{3 y}{2}) = - \frac{2}{3} (x - \frac{7}{2})$

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

$\frac{- 2}{3} \cdot \frac{- 3 y}{2} = - \frac{2}{3} (x - \frac{7}{2})$

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 y}{2} = - \frac{2}{3} (x - \frac{7}{2})$

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 y}{2} = - \frac{2}{3} (x - \frac{7}{2})$

Rewrite the expression.

$\frac{- 1}{3} (- 3 y) = - \frac{2}{3} (x - \frac{7}{2})$

Cancel the common factor of $3$ .

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

$\frac{- 1}{3} (3 (- y)) = - \frac{2}{3} (x - \frac{7}{2})$

Cancel the common factor.

$\frac{- 1}{3} (3 (- y)) = - \frac{2}{3} (x - \frac{7}{2})$

Rewrite the expression.

$- - y = - \frac{2}{3} (x - \frac{7}{2})$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{2}{3} (x - \frac{7}{2})$

Multiply $y$ by $1$ .

$y = - \frac{2}{3} (x - \frac{7}{2})$

Simplify the right side.

Simplify $- \frac{2}{3} (x - \frac{7}{2})$ .

Simplify terms.

Apply the distributive property.

$y = - \frac{2}{3} x - \frac{2}{3} (- \frac{7}{2})$

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

$y = - \frac{x \cdot 2}{3} - \frac{2}{3} (- \frac{7}{2})$

Cancel the common factor of $2$ .

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

$y = - \frac{x \cdot 2}{3} + \frac{- 2}{3} (- \frac{7}{2})$

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$y = - \frac{x \cdot 2}{3} + \frac{- 2}{3} \cdot \frac{- 7}{2}$

Factor $2$ out of $- 2$ .

$y = - \frac{x \cdot 2}{3} + \frac{2 (- 1)}{3} \cdot \frac{- 7}{2}$

Cancel the common factor.

$y = - \frac{x \cdot 2}{3} + \frac{2 \cdot - 1}{3} \cdot \frac{- 7}{2}$

Rewrite the expression.

$y = - \frac{x \cdot 2}{3} + \frac{- 1}{3} \cdot - 7$

Combine $\frac{- 1}{3}$ and $- 7$ .

$y = - \frac{x \cdot 2}{3} + \frac{- - 7}{3}$

Multiply $- 1$ by $- 7$ .

$y = - \frac{x \cdot 2}{3} + \frac{7}{3}$

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

$y = - \frac{2 x}{3} + \frac{7}{3}$

Step 4

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

$F^{- 1} (x) = - \frac{2 x}{3} + \frac{7}{3}$

Step 5

Verify if $F^{- 1} (x) = - \frac{2 x}{3} + \frac{7}{3}$ is the inverse of $F (x) = - \frac{3}{2} x + \frac{7}{2}$ .

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 $F^{- 1} (- \frac{3}{2} x + \frac{7}{2})$ by substituting in the value of $F$ into $F^{- 1}$ .

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = - \frac{2 (- \frac{3}{2} x + \frac{7}{2})}{3} + \frac{7}{3}$

Combine the numerators over the common denominator.

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{- 2 (- \frac{3}{2} x + \frac{7}{2}) + 7}{3}$

Simplify each term.

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

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{- 2 (- \frac{x \cdot 3}{2} + \frac{7}{2}) + 7}{3}$

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

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{- 2 (- \frac{3 x}{2} + \frac{7}{2}) + 7}{3}$

Apply the distributive property.

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{- 2 (- \frac{3 x}{2}) - 2 (\frac{7}{2}) + 7}{3}$

Cancel the common factor of $2$ .

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

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{- 2 \frac{- 3 x}{2} - 2 (\frac{7}{2}) + 7}{3}$

Factor $2$ out of $- 2$ .

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{2 (- 1) (\frac{- 3 x}{2}) - 2 (\frac{7}{2}) + 7}{3}$

Cancel the common factor.

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{2 \cdot (- 1 \frac{- 3 x}{2}) - 2 (\frac{7}{2}) + 7}{3}$

Rewrite the expression.

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{- 1 (- 3 x) - 2 (\frac{7}{2}) + 7}{3}$

Multiply $- 3$ by $- 1$ .

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{3 x - 2 (\frac{7}{2}) + 7}{3}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{3 x + 2 (- 1) (\frac{7}{2}) + 7}{3}$

Cancel the common factor.

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{3 x + 2 \cdot (- 1 (\frac{7}{2})) + 7}{3}$

Rewrite the expression.

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{3 x - 1 \cdot 7 + 7}{3}$

Multiply $- 1$ by $7$ .

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{3 x - 7 + 7}{3}$

Simplify terms.

Combine the opposite terms in $3 x - 7 + 7$ .

Add $- 7$ and $7$ .

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{3 x + 0}{3}$

Add $3 x$ and $0$ .

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{3 x}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = \frac{3 x}{3}$

Divide $x$ by $1$ .

$F^{- 1} (- \frac{3}{2} x + \frac{7}{2}) = x$

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

Set up the composite result function.

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

Evaluate $F (- \frac{2 x}{3} + \frac{7}{3})$ by substituting in the value of $F^{- 1}$ into $F$ .

$F (- \frac{2 x}{3} + \frac{7}{3}) = - \frac{3}{2} \cdot (- \frac{2 x}{3} + \frac{7}{3}) + \frac{7}{2}$

Simplify each term.

Apply the distributive property.

$F (- \frac{2 x}{3} + \frac{7}{3}) = - \frac{3}{2} \cdot (- \frac{2 x}{3}) - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Cancel the common factor of $3$ .

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

$F (- \frac{2 x}{3} + \frac{7}{3}) = \frac{- 3}{2} \cdot (- \frac{2 x}{3}) - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

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

$F (- \frac{2 x}{3} + \frac{7}{3}) = \frac{- 3}{2} \cdot \frac{- 2 x}{3} - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Factor $3$ out of $- 3$ .

$F (- \frac{2 x}{3} + \frac{7}{3}) = \frac{3 (- 1)}{2} \cdot \frac{- 2 x}{3} - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Cancel the common factor.

$F (- \frac{2 x}{3} + \frac{7}{3}) = \frac{3 \cdot - 1}{2} \cdot \frac{- 2 x}{3} - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Rewrite the expression.

$F (- \frac{2 x}{3} + \frac{7}{3}) = \frac{- 1}{2} \cdot (- 2 x) - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2 x$ .

$F (- \frac{2 x}{3} + \frac{7}{3}) = \frac{- 1}{2} \cdot (2 (- x)) - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Cancel the common factor.

$F (- \frac{2 x}{3} + \frac{7}{3}) = \frac{- 1}{2} \cdot (2 (- x)) - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Rewrite the expression.

$F (- \frac{2 x}{3} + \frac{7}{3}) = - 1 (- x) - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Multiply $- 1$ by $- 1$ .

$F (- \frac{2 x}{3} + \frac{7}{3}) = 1 x - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Multiply $x$ by $1$ .

$F (- \frac{2 x}{3} + \frac{7}{3}) = x - \frac{3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Cancel the common factor of $3$ .

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

$F (- \frac{2 x}{3} + \frac{7}{3}) = x + \frac{- 3}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Factor $3$ out of $- 3$ .

$F (- \frac{2 x}{3} + \frac{7}{3}) = x + \frac{3 (- 1)}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Cancel the common factor.

$F (- \frac{2 x}{3} + \frac{7}{3}) = x + \frac{3 \cdot - 1}{2} \cdot \frac{7}{3} + \frac{7}{2}$

Rewrite the expression.

$F (- \frac{2 x}{3} + \frac{7}{3}) = x + \frac{- 1}{2} \cdot 7 + \frac{7}{2}$

Combine $\frac{- 1}{2}$ and $7$ .

$F (- \frac{2 x}{3} + \frac{7}{3}) = x + \frac{- 1 \cdot 7}{2} + \frac{7}{2}$

Multiply $- 1$ by $7$ .

$F (- \frac{2 x}{3} + \frac{7}{3}) = x + \frac{- 7}{2} + \frac{7}{2}$

Move the negative in front of the fraction.

$F (- \frac{2 x}{3} + \frac{7}{3}) = x - \frac{7}{2} + \frac{7}{2}$

Combine the opposite terms in $x - \frac{7}{2} + \frac{7}{2}$ .

Add $- \frac{7}{2}$ and $\frac{7}{2}$ .

$F (- \frac{2 x}{3} + \frac{7}{3}) = x + 0$

Add $x$ and $0$ .

$F (- \frac{2 x}{3} + \frac{7}{3}) = x$

Since $F^{- 1} (F (x)) = x$ and $F (F^{- 1} (x)) = x$ , then $F^{- 1} (x) = - \frac{2 x}{3} + \frac{7}{3}$ is the inverse of $F (x) = - \frac{3}{2} x + \frac{7}{2}$ .

$F^{- 1} (x) = - \frac{2 x}{3} + \frac{7}{3}$

Find the Inverse F(x)=-3/2x+7/2

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