Find the Inverse 6/(x-4)

$\frac{6}{x - 4}$

Step 1

Interchange the variables.

$x = \frac{6}{y - 4}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{6}{y - 4} = x$ .

$\frac{6}{y - 4} = x$

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.

$y - 4, 1$

Remove parentheses.

$y - 4, 1$

The LCM of one and any expression is the expression.

$y - 4$

Multiply each term in $\frac{6}{y - 4} = x$ by $y - 4$ to eliminate the fractions.

Multiply each term in $\frac{6}{y - 4} = x$ by $y - 4$ .

$\frac{6}{y - 4} (y - 4) = x (y - 4)$

Simplify the left side.

Cancel the common factor of $y - 4$ .

Cancel the common factor.

$\frac{6}{y - 4} (y - 4) = x (y - 4)$

Rewrite the expression.

$6 = x (y - 4)$

Simplify the right side.

Apply the distributive property.

$6 = x y + x \cdot - 4$

Move $- 4$ to the left of $x$ .

$6 = x y - 4 x$

Solve the equation.

Rewrite the equation as $x y - 4 x = 6$ .

$x y - 4 x = 6$

Add $4 x$ to both sides of the equation.

$x y = 6 + 4 x$

Divide each term in $x y = 6 + 4 x$ by $x$ and simplify.

Divide each term in $x y = 6 + 4 x$ by $x$ .

$\frac{x y}{x} = \frac{6}{x} + \frac{4 x}{x}$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x y}{x} = \frac{6}{x} + \frac{4 x}{x}$

Divide $y$ by $1$ .

$y = \frac{6}{x} + \frac{4 x}{x}$

Simplify the right side.

Cancel the common factor of $x$ .

Cancel the common factor.

$y = \frac{6}{x} + \frac{4 x}{x}$

Divide $4$ by $1$ .

$y = \frac{6}{x} + 4$

Step 3

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

$f^{- 1} (x) = \frac{6}{x} + 4$

Step 4

Verify if $f^{- 1} (x) = \frac{6}{x} + 4$ is the inverse of $f (x) = \frac{6}{x - 4}$ .

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 $f^{- 1} (\frac{6}{x - 4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6}{x - 4}) = \frac{6}{\frac{6}{x - 4}} + 4$

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6}{x - 4}) = 6 (\frac{x - 4}{6}) + 4$

Cancel the common factor of $6$ .

Cancel the common factor.

$f^{- 1} (\frac{6}{x - 4}) = 6 (\frac{x - 4}{6}) + 4$

Rewrite the expression.

$f^{- 1} (\frac{6}{x - 4}) = x - 4 + 4$

Combine the opposite terms in $x - 4 + 4$ .

Add $- 4$ and $4$ .

$f^{- 1} (\frac{6}{x - 4}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{6}{x - 4}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{6}{x} + 4)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6}{x} + 4) = \frac{6}{(\frac{6}{x} + 4) - 4}$

Cancel the common factor of $6$ and $(\frac{6}{x} + 4) - 4$ .

Factor $2$ out of $6$ .

$f (\frac{6}{x} + 4) = \frac{2 (3)}{(\frac{6}{x} + 4) - 4}$

Cancel the common factors.

Factor $2$ out of $\frac{6}{x}$ .

$f (\frac{6}{x} + 4) = \frac{2 \cdot 3}{2 (\frac{3}{x}) + 4 - 4}$

Factor $2$ out of $4$ .

$f (\frac{6}{x} + 4) = \frac{2 (3)}{2 (\frac{3}{x}) + 2 \cdot 2 - 4}$

Factor $2$ out of $2 \frac{3}{x} + 2 \cdot 2$ .

$f (\frac{6}{x} + 4) = \frac{2 (3)}{2 (\frac{3}{x} + 2) - 4}$

Factor $2$ out of $- 4$ .

$f (\frac{6}{x} + 4) = \frac{2 \cdot 3}{2 (\frac{3}{x} + 2) + 2 \cdot - 2}$

Factor $2$ out of $2 (\frac{3}{x} + 2) + 2 (- 2)$ .

$f (\frac{6}{x} + 4) = \frac{2 (3)}{2 (\frac{3}{x} + 2 - 2)}$

Cancel the common factor.

$f (\frac{6}{x} + 4) = \frac{2 \cdot 3}{2 (\frac{3}{x} + 2 - 2)}$

Rewrite the expression.

$f (\frac{6}{x} + 4) = \frac{3}{\frac{3}{x} + 2 - 2}$

Simplify the denominator.

Subtract $2$ from $2$ .

$f (\frac{6}{x} + 4) = \frac{3}{\frac{3}{x} + 0}$

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

$f (\frac{6}{x} + 4) = \frac{3}{\frac{3}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{6}{x} + 4) = 3 (\frac{x}{3})$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{6}{x} + 4) = 3 (\frac{x}{3})$

Rewrite the expression.

$f (\frac{6}{x} + 4) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6}{x} + 4$ is the inverse of $f (x) = \frac{6}{x - 4}$ .

$f^{- 1} (x) = \frac{6}{x} + 4$

Find the Inverse 6/(x-4)

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