Find the Inverse 4/(11-2x)

$\frac{4}{11 - 2 x}$

Step 1

Interchange the variables.

$x = \frac{4}{11 - 2 y}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{4}{11 - 2 y} = x$ .

$\frac{4}{11 - 2 y} = x$

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.

$11 - 2 y, 1$

Remove parentheses.

$11 - 2 y, 1$

The LCM of one and any expression is the expression.

$11 - 2 y$

Multiply each term in $\frac{4}{11 - 2 y} = x$ by $11 - 2 y$ to eliminate the fractions.

Multiply each term in $\frac{4}{11 - 2 y} = x$ by $11 - 2 y$ .

$\frac{4}{11 - 2 y} (11 - 2 y) = x (11 - 2 y)$

Simplify the left side.

Cancel the common factor of $11 - 2 y$ .

Cancel the common factor.

$\frac{4}{11 - 2 y} (11 - 2 y) = x (11 - 2 y)$

Rewrite the expression.

$4 = x (11 - 2 y)$

Simplify the right side.

Apply the distributive property.

$4 = x \cdot 11 + x (- 2 y)$

Reorder.

Move $11$ to the left of $x$ .

$4 = 11 \cdot x + x (- 2 y)$

Rewrite using the commutative property of multiplication.

$4 = 11 x - 2 x y$

Solve the equation.

Rewrite the equation as $11 x - 2 x y = 4$ .

$11 x - 2 x y = 4$

Subtract $11 x$ from both sides of the equation.

$- 2 x y = 4 - 11 x$

Divide each term in $- 2 x y = 4 - 11 x$ by $- 2 x$ and simplify.

Divide each term in $- 2 x y = 4 - 11 x$ by $- 2 x$ .

$\frac{- 2 x y}{- 2 x} = \frac{4}{- 2 x} + \frac{- 11 x}{- 2 x}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x y}{- 2 x} = \frac{4}{- 2 x} + \frac{- 11 x}{- 2 x}$

Rewrite the expression.

$\frac{x y}{x} = \frac{4}{- 2 x} + \frac{- 11 x}{- 2 x}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x y}{x} = \frac{4}{- 2 x} + \frac{- 11 x}{- 2 x}$

Divide $y$ by $1$ .

$y = \frac{4}{- 2 x} + \frac{- 11 x}{- 2 x}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $4$ and $- 2$ .

Factor $2$ out of $4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Dividing two negative values results in a positive value.

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

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = - \frac{2}{x} + \frac{11}{2}$ is the inverse of $f (x) = \frac{4}{11 - 2 x}$ .

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 $f^{- 1} (\frac{4}{11 - 2 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = - \frac{2}{\frac{4}{11 - 2 x}} + \frac{11}{2}$

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{4}{11 - 2 x}) = - (2 (\frac{11 - 2 x}{4})) + \frac{11}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = - (2 (\frac{11 - 2 x}{2 (2)})) + \frac{11}{2}$

Cancel the common factor.

$f^{- 1} (\frac{4}{11 - 2 x}) = - (2 (\frac{11 - 2 x}{2 \cdot 2})) + \frac{11}{2}$

Rewrite the expression.

$f^{- 1} (\frac{4}{11 - 2 x}) = - \frac{11 - 2 x}{2} + \frac{11}{2}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{- (11 - 2 x) + 11}{2}$

Simplify each term.

Apply the distributive property.

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{- 1 \cdot 11 - (- 2 x) + 11}{2}$

Multiply $- 1$ by $11$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{- 11 - (- 2 x) + 11}{2}$

Multiply $- 2$ by $- 1$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{- 11 + 2 x + 11}{2}$

Simplify terms.

Combine the opposite terms in $- 11 + 2 x + 11$ .

Add $- 11$ and $11$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{2 x + 0}{2}$

Add $2 x$ and $0$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{2 x}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\frac{4}{11 - 2 x}) = \frac{2 x}{2}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{4}{11 - 2 x}) = x$

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

Set up the composite result function.

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

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

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 - 2 (- \frac{2}{x} + \frac{11}{2})}$

Simplify the denominator.

Apply the distributive property.

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 - 2 (- \frac{2}{x}) - 2 (\frac{11}{2})}$

Multiply $- 2 (- \frac{2}{x})$ .

Multiply $- 1$ by $- 2$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + 2 (\frac{2}{x}) - 2 (\frac{11}{2})}$

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

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{2 \cdot 2}{x} - 2 (\frac{11}{2})}$

Multiply $2$ by $2$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{4}{x} - 2 (\frac{11}{2})}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{4}{x} + 2 (- 1) (\frac{11}{2})}$

Cancel the common factor.

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{4}{x} + 2 \cdot (- 1 (\frac{11}{2}))}$

Rewrite the expression.

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{4}{x} - 1 \cdot 11}$

Multiply $- 1$ by $11$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{11 + \frac{4}{x} - 11}$

Subtract $11$ from $11$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{\frac{4}{x} + 0}$

Add $\frac{4}{x}$ and $0$ .

$f (- \frac{2}{x} + \frac{11}{2}) = \frac{4}{\frac{4}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{2}{x} + \frac{11}{2}) = 4 (\frac{x}{4})$

Cancel the common factor of $4$ .

Cancel the common factor.

$f (- \frac{2}{x} + \frac{11}{2}) = 4 (\frac{x}{4})$

Rewrite the expression.

$f (- \frac{2}{x} + \frac{11}{2}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{2}{x} + \frac{11}{2}$ is the inverse of $f (x) = \frac{4}{11 - 2 x}$ .

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

Find the Inverse 4/(11-2x)

Related Questions

Support LCMGCF.com by disabling your adblocker.