Find the Inverse F(x)=4/(2-x)

$F (x) = \frac{4}{2 - x}$

Step 1

Write $F (x) = \frac{4}{2 - x}$ as an equation.

$y = \frac{4}{2 - x}$

Step 2

Interchange the variables.

$x = \frac{4}{2 - y}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{4}{2 - y} = x$ .

$\frac{4}{2 - y} = x$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 - y, 1$

Remove parentheses.

$2 - y, 1$

The LCM of one and any expression is the expression.

$2 - y$

Multiply each term in $\frac{4}{2 - y} = x$ by $2 - y$ to eliminate the fractions.

Multiply each term in $\frac{4}{2 - y} = x$ by $2 - y$ .

$\frac{4}{2 - y} (2 - y) = x (2 - y)$

Simplify the left side.

Cancel the common factor of $2 - y$ .

Cancel the common factor.

$\frac{4}{2 - y} (2 - y) = x (2 - y)$

Rewrite the expression.

$4 = x (2 - y)$

Simplify the right side.

Apply the distributive property.

$4 = x \cdot 2 + x (- y)$

Reorder.

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

$4 = 2 \cdot x + x (- y)$

Rewrite using the commutative property of multiplication.

$4 = 2 x - x y$

Solve the equation.

Rewrite the equation as $2 x - x y = 4$ .

$2 x - x y = 4$

Subtract $2 x$ from both sides of the equation.

$- x y = 4 - 2 x$

Divide each term in $- x y = 4 - 2 x$ by $- 1 x$ and simplify.

Divide each term in $- x y = 4 - 2 x$ by $- 1 x$ .

$\frac{- x y}{- 1 x} = \frac{4}{- 1 x} + \frac{- 2 x}{- 1 x}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x y}{x} = \frac{4}{- 1 x} + \frac{- 2 x}{- 1 x}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x y}{x} = \frac{4}{- 1 x} + \frac{- 2 x}{- 1 x}$

Divide $y$ by $1$ .

$y = \frac{4}{- 1 x} + \frac{- 2 x}{- 1 x}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{4}{x} + \frac{- 2 x}{- 1 x}$

Cancel the common factor of $x$ .

Cancel the common factor.

$y = - \frac{4}{x} + \frac{- 2 x}{- 1 x}$

Rewrite the expression.

$y = - \frac{4}{x} + \frac{- 2}{- 1}$

Move the negative one from the denominator of $\frac{- 2}{- 1}$ .

$y = - \frac{4}{x} - 1 \cdot - 2$

Rewrite $- 1 \cdot - 2$ as $- - 2$ .

$y = - \frac{4}{x} - - 2$

Multiply $- 1$ by $- 2$ .

$y = - \frac{4}{x} + 2$

Step 4

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

$F^{- 1} (x) = - \frac{4}{x} + 2$

Step 5

Verify if $F^{- 1} (x) = - \frac{4}{x} + 2$ is the inverse of $F (x) = \frac{4}{2 - x}$ .

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 $F^{- 1} (\frac{4}{2 - x})$ by substituting in the value of $F$ into $F^{- 1}$ .

$F^{- 1} (\frac{4}{2 - x}) = - \frac{4}{\frac{4}{2 - x}} + 2$

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$F^{- 1} (\frac{4}{2 - x}) = - (4 (\frac{2 - x}{4})) + 2$

Cancel the common factor of $4$ .

Cancel the common factor.

$F^{- 1} (\frac{4}{2 - x}) = - (4 (\frac{2 - x}{4})) + 2$

Rewrite the expression.

$F^{- 1} (\frac{4}{2 - x}) = - (2 - x) + 2$

Apply the distributive property.

$F^{- 1} (\frac{4}{2 - x}) = - 1 \cdot 2 + x + 2$

Multiply $- 1$ by $2$ .

$F^{- 1} (\frac{4}{2 - x}) = - 2 + x + 2$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$F^{- 1} (\frac{4}{2 - x}) = - 2 + 1 x + 2$

Multiply $x$ by $1$ .

$F^{- 1} (\frac{4}{2 - x}) = - 2 + x + 2$

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

Add $- 2$ and $2$ .

$F^{- 1} (\frac{4}{2 - x}) = x + 0$

Add $x$ and $0$ .

$F^{- 1} (\frac{4}{2 - x}) = x$

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

Set up the composite result function.

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

Evaluate $F (- \frac{4}{x} + 2)$ by substituting in the value of $F^{- 1}$ into $F$ .

$F (- \frac{4}{x} + 2) = \frac{4}{2 - (- \frac{4}{x} + 2)}$

Cancel the common factor of $4$ and $2 - (- \frac{4}{x} + 2)$ .

Rewrite $2$ as $- 1 (- 2)$ .

$F (- \frac{4}{x} + 2) = \frac{4}{- 1 \cdot - 2 - (- \frac{4}{x} + 2)}$

Factor $- 1$ out of $- 1 (- 2) - (- \frac{4}{x} + 2)$ .

$F (- \frac{4}{x} + 2) = \frac{4}{- 1 (- 2 - \frac{4}{x} + 2)}$

Factor $2$ out of $4$ .

$F (- \frac{4}{x} + 2) = \frac{2 \cdot 2}{- 1 (- 2 - \frac{4}{x} + 2)}$

Cancel the common factors.

Factor $2$ out of $- 1 (- 2 - \frac{4}{x} + 2)$ .

$F (- \frac{4}{x} + 2) = \frac{2 \cdot 2}{2 (- 1 (- 1 - \frac{2}{x} + 1))}$

Cancel the common factor.

$F (- \frac{4}{x} + 2) = \frac{2 \cdot 2}{2 (- 1 (- 1 - \frac{2}{x} + 1))}$

Rewrite the expression.

$F (- \frac{4}{x} + 2) = \frac{2}{- 1 (- 1 - \frac{2}{x} + 1)}$

Simplify the denominator.

Add $- 1$ and $1$ .

$F (- \frac{4}{x} + 2) = \frac{2}{- 1 (- \frac{2}{x} + 0)}$

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

$F (- \frac{4}{x} + 2) = \frac{2}{- 1 \cdot (- 1 \frac{2}{x})}$

Combine exponents.

Factor out negative.

$F (- \frac{4}{x} + 2) = \frac{2}{\frac{2}{x}}$

Multiply $- 1$ by $- 1$ .

$F (- \frac{4}{x} + 2) = \frac{2}{1 (\frac{2}{x})}$

Multiply $\frac{2}{x}$ by $1$ .

$F (- \frac{4}{x} + 2) = \frac{2}{\frac{2}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$F (- \frac{4}{x} + 2) = 2 (\frac{x}{2})$

Cancel the common factor of $2$ .

Cancel the common factor.

$F (- \frac{4}{x} + 2) = 2 (\frac{x}{2})$

Rewrite the expression.

$F (- \frac{4}{x} + 2) = x$

Since $F^{- 1} (F (x)) = x$ and $F (F^{- 1} (x)) = x$ , then $F^{- 1} (x) = - \frac{4}{x} + 2$ is the inverse of $F (x) = \frac{4}{2 - x}$ .

$F^{- 1} (x) = - \frac{4}{x} + 2$

Find the Inverse F(x)=4/(2-x)

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