Find the Inverse (7x)/(x+8)

$\frac{7 x}{x + 8}$

Step 1

Interchange the variables.

$x = \frac{7 y}{y + 8}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{7 y}{y + 8} = x$ .

$\frac{7 y}{y + 8} = x$

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.

$y + 8, 1$

Remove parentheses.

$y + 8, 1$

The LCM of one and any expression is the expression.

$y + 8$

Multiply each term in $\frac{7 y}{y + 8} = x$ by $y + 8$ to eliminate the fractions.

Multiply each term in $\frac{7 y}{y + 8} = x$ by $y + 8$ .

$\frac{7 y}{y + 8} (y + 8) = x (y + 8)$

Simplify the left side.

Cancel the common factor of $y + 8$ .

Cancel the common factor.

$\frac{7 y}{y + 8} (y + 8) = x (y + 8)$

Rewrite the expression.

$7 y = x (y + 8)$

Simplify the right side.

Apply the distributive property.

$7 y = x y + x \cdot 8$

Move $8$ to the left of $x$ .

$7 y = x y + 8 x$

Solve the equation.

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

$7 y - x y = 8 x$

Factor $y$ out of $7 y - x y$ .

Factor $y$ out of $7 y$ .

$y \cdot 7 - x y = 8 x$

Factor $y$ out of $- x y$ .

$y \cdot 7 + y (- x) = 8 x$

Factor $y$ out of $y \cdot 7 + y (- x)$ .

$y (7 - x) = 8 x$

Divide each term in $y (7 - x) = 8 x$ by $7 - x$ and simplify.

Divide each term in $y (7 - x) = 8 x$ by $7 - x$ .

$\frac{y (7 - x)}{7 - x} = \frac{8 x}{7 - x}$

Simplify the left side.

Cancel the common factor of $7 - x$ .

Cancel the common factor.

$\frac{y (7 - x)}{7 - x} = \frac{8 x}{7 - x}$

Divide $y$ by $1$ .

$y = \frac{8 x}{7 - x}$

Step 3

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

$f^{- 1} (x) = \frac{8 x}{7 - x}$

Step 4

Verify if $f^{- 1} (x) = \frac{8 x}{7 - x}$ is the inverse of $f (x) = \frac{7 x}{x + 8}$ .

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 $f^{- 1} (\frac{7 x}{x + 8})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{8 (\frac{7 x}{x + 8})}{7 - (\frac{7 x}{x + 8})}$

Combine $8$ and $\frac{7 x}{x + 8}$ .

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{\frac{8 (7 x)}{x + 8}}{7 - \frac{7 x}{x + 8}}$

Simplify the denominator.

To write $7$ as a fraction with a common denominator, multiply by $\frac{x + 8}{x + 8}$ .

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{\frac{8 (7 x)}{x + 8}}{\frac{7 (x + 8)}{x + 8} - \frac{7 x}{x + 8}}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{\frac{8 (7 x)}{x + 8}}{\frac{7 (x + 8) - 7 x}{x + 8}}$

Reorder terms.

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{\frac{8 (7 x)}{x + 8}}{\frac{- 7 x + 7 (x + 8)}{x + 8}}$

Rewrite $\frac{- 7 x + 7 (x + 8)}{x + 8}$ in a factored form.

Factor $7$ out of $- 7 x + 7 (x + 8)$ .

Factor $7$ out of $- 7 x$ .

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{\frac{8 (7 x)}{x + 8}}{\frac{7 (- x) + 7 (x + 8)}{x + 8}}$

Factor $7$ out of $7 (- x) + 7 (x + 8)$ .

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{\frac{8 (7 x)}{x + 8}}{\frac{7 (- x + x + 8)}{x + 8}}$

Add $- x$ and $x$ .

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{\frac{8 (7 x)}{x + 8}}{\frac{7 (0 + 8)}{x + 8}}$

Add $0$ and $8$ .

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{\frac{8 (7 x)}{x + 8}}{\frac{7 \cdot 8}{x + 8}}$

Multiply $7$ by $8$ .

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{\frac{8 (7 x)}{x + 8}}{\frac{56}{x + 8}}$

Multiply $8$ by $7$ .

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{\frac{56 x}{x + 8}}{\frac{56}{x + 8}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{56 x}{x + 8} \cdot \frac{x + 8}{56}$

Cancel the common factor of $56$ .

Factor $56$ out of $56 x$ .

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{56 (x)}{x + 8} \cdot \frac{x + 8}{56}$

Cancel the common factor.

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{56 x}{x + 8} \cdot \frac{x + 8}{56}$

Rewrite the expression.

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{x}{x + 8} \cdot (x + 8)$

Cancel the common factor of $x + 8$ .

Cancel the common factor.

$f^{- 1} (\frac{7 x}{x + 8}) = \frac{x}{x + 8} \cdot (x + 8)$

Rewrite the expression.

$f^{- 1} (\frac{7 x}{x + 8}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{8 x}{7 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{8 x}{7 - x}) = \frac{7 (\frac{8 x}{7 - x})}{(\frac{8 x}{7 - x}) + 8}$

Combine $7$ and $\frac{8 x}{7 - x}$ .

$f (\frac{8 x}{7 - x}) = \frac{\frac{7 (8 x)}{7 - x}}{\frac{8 x}{7 - x} + 8}$

Simplify the denominator.

To write $8$ as a fraction with a common denominator, multiply by $\frac{7 - x}{7 - x}$ .

$f (\frac{8 x}{7 - x}) = \frac{\frac{7 (8 x)}{7 - x}}{\frac{8 x}{7 - x} + \frac{8 (7 - x)}{7 - x}}$

Combine the numerators over the common denominator.

$f (\frac{8 x}{7 - x}) = \frac{\frac{7 (8 x)}{7 - x}}{\frac{8 x + 8 (7 - x)}{7 - x}}$

Reorder terms.

$f (\frac{8 x}{7 - x}) = \frac{\frac{7 (8 x)}{7 - x}}{\frac{8 x + 8 (7 - x)}{- x + 7}}$

Rewrite $\frac{8 x + 8 (7 - x)}{- x + 7}$ in a factored form.

Factor $8$ out of $8 x + 8 (7 - x)$ .

Factor $8$ out of $8 x$ .

$f (\frac{8 x}{7 - x}) = \frac{\frac{7 (8 x)}{7 - x}}{\frac{8 (x) + 8 (7 - x)}{- x + 7}}$

Factor $8$ out of $8 (x) + 8 (7 - x)$ .

$f (\frac{8 x}{7 - x}) = \frac{\frac{7 (8 x)}{7 - x}}{\frac{8 (x + 7 - x)}{- x + 7}}$

Subtract $x$ from $x$ .

$f (\frac{8 x}{7 - x}) = \frac{\frac{7 (8 x)}{7 - x}}{\frac{8 (0 + 7)}{- x + 7}}$

Add $0$ and $7$ .

$f (\frac{8 x}{7 - x}) = \frac{\frac{7 (8 x)}{7 - x}}{\frac{8 \cdot 7}{- x + 7}}$

Multiply $8$ by $7$ .

$f (\frac{8 x}{7 - x}) = \frac{\frac{7 (8 x)}{7 - x}}{\frac{56}{- x + 7}}$

Multiply $7$ by $8$ .

$f (\frac{8 x}{7 - x}) = \frac{\frac{56 x}{7 - x}}{\frac{56}{- x + 7}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{8 x}{7 - x}) = \frac{56 x}{7 - x} \cdot \frac{- x + 7}{56}$

Cancel the common factor of $56$ .

Factor $56$ out of $56 x$ .

$f (\frac{8 x}{7 - x}) = \frac{56 (x)}{7 - x} \cdot \frac{- x + 7}{56}$

Cancel the common factor.

$f (\frac{8 x}{7 - x}) = \frac{56 x}{7 - x} \cdot \frac{- x + 7}{56}$

Rewrite the expression.

$f (\frac{8 x}{7 - x}) = \frac{x}{7 - x} \cdot (- x + 7)$

Multiply $\frac{x}{7 - x}$ by $- x + 7$ .

$f (\frac{8 x}{7 - x}) = \frac{x (- x + 7)}{7 - x}$

Cancel the common factor of $- x + 7$ and $7 - x$ .

Reorder terms.

$f (\frac{8 x}{7 - x}) = \frac{x (- x + 7)}{- x + 7}$

Cancel the common factor.

$f (\frac{8 x}{7 - x}) = \frac{x (- x + 7)}{- x + 7}$

Divide $x$ by $1$ .

$f (\frac{8 x}{7 - x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{8 x}{7 - x}$ is the inverse of $f (x) = \frac{7 x}{x + 8}$ .

$f^{- 1} (x) = \frac{8 x}{7 - x}$

Find the Inverse (7x)/(x+8)

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