Find the Inverse 6(4/x)-12

$6 (\frac{4}{x}) - 12$

Step 1

Interchange the variables.

$x = 6 (\frac{4}{y}) - 12$

Step 2

Solve for $y$ .

Rewrite the equation as $6 (\frac{4}{y}) - 12 = x$ .

$6 (\frac{4}{y}) - 12 = x$

Multiply $6 (\frac{4}{y})$ .

Combine $6$ and $\frac{4}{y}$ .

$\frac{6 \cdot 4}{y} - 12 = x$

Multiply $6$ by $4$ .

$\frac{24}{y} - 12 = x$

Add $12$ to both sides of the equation.

$\frac{24}{y} = x + 12$

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.

$y, 1, 1$

The LCM of one and any expression is the expression.

$y$

Multiply each term in $\frac{24}{y} = x + 12$ by $y$ to eliminate the fractions.

Multiply each term in $\frac{24}{y} = x + 12$ by $y$ .

$\frac{24}{y} y = x y + 12 y$

Simplify the left side.

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{24}{y} y = x y + 12 y$

Rewrite the expression.

$24 = x y + 12 y$

Solve the equation.

Rewrite the equation as $x y + 12 y = 24$ .

$x y + 12 y = 24$

Factor $y$ out of $x y + 12 y$ .

Factor $y$ out of $x y$ .

$y x + 12 y = 24$

Factor $y$ out of $12 y$ .

$y x + y \cdot 12 = 24$

Factor $y$ out of $y x + y \cdot 12$ .

$y (x + 12) = 24$

Divide each term in $y (x + 12) = 24$ by $x + 12$ and simplify.

Divide each term in $y (x + 12) = 24$ by $x + 12$ .

$\frac{y (x + 12)}{x + 12} = \frac{24}{x + 12}$

Simplify the left side.

Cancel the common factor of $x + 12$ .

Cancel the common factor.

$\frac{y (x + 12)}{x + 12} = \frac{24}{x + 12}$

Divide $y$ by $1$ .

$y = \frac{24}{x + 12}$

Step 3

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

$f^{- 1} (x) = \frac{24}{x + 12}$

Step 4

Verify if $f^{- 1} (x) = \frac{24}{x + 12}$ is the inverse of $f (x) = 6 (\frac{4}{x}) - 12$ .

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 $f^{- 1} (6 (\frac{4}{x}) - 12)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{24}{(6 (\frac{4}{x}) - 12) + 12}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $24$ and $(6 (\frac{4}{x}) - 12) + 12$ .

Factor $6$ out of $24$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{6 \cdot 4}{6 (\frac{4}{x}) - 12 + 12}$

Cancel the common factors.

Factor $6$ out of $- 12$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{6 \cdot 4}{6 (\frac{4}{x}) + 6 (- 2) + 12}$

Factor $6$ out of $6 (\frac{4}{x}) + 6 (- 2)$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{6 \cdot 4}{6 (\frac{4}{x} - 2) + 12}$

Factor $6$ out of $12$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{6 \cdot 4}{6 (\frac{4}{x} - 2) + 6 (2)}$

Factor $6$ out of $6 (\frac{4}{x} - 2) + 6 (2)$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{6 \cdot 4}{6 (\frac{4}{x} - 2 + 2)}$

Cancel the common factor.

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{6 \cdot 4}{6 (\frac{4}{x} - 2 + 2)}$

Rewrite the expression.

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{4}{\frac{4}{x} - 2 + 2}$

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

Factor $2$ out of $4$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{2 \cdot 2}{\frac{4}{x} - 2 + 2}$

Cancel the common factors.

Factor $2$ out of $\frac{4}{x}$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{2 \cdot 2}{2 (\frac{2}{x}) - 2 + 2}$

Factor $2$ out of $- 2$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{2 \cdot 2}{2 (\frac{2}{x}) + 2 \cdot - 1 + 2}$

Factor $2$ out of $2 \frac{2}{x} + 2 \cdot - 1$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{2 \cdot 2}{2 (\frac{2}{x} - 1) + 2}$

Factor $2$ out of $2$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{2 \cdot 2}{2 (\frac{2}{x} - 1) + 2 (1)}$

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

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{2 \cdot 2}{2 (\frac{2}{x} - 1 + 1)}$

Cancel the common factor.

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{2 \cdot 2}{2 (\frac{2}{x} - 1 + 1)}$

Rewrite the expression.

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{2}{\frac{2}{x} - 1 + 1}$

Simplify the denominator.

Add $- 1$ and $1$ .

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{2}{\frac{2}{x} + 0}$

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

$f^{- 1} (6 (\frac{4}{x}) - 12) = \frac{2}{\frac{2}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (6 (\frac{4}{x}) - 12) = 2 (\frac{x}{2})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (6 (\frac{4}{x}) - 12) = 2 (\frac{x}{2})$

Rewrite the expression.

$f^{- 1} (6 (\frac{4}{x}) - 12) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{24}{x + 12})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{24}{x + 12}) = 6 (\frac{4}{\frac{24}{x + 12}}) - 12$

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{24}{x + 12}) = 6 (4 (\frac{x + 12}{24})) - 12$

Cancel the common factor of $4$ .

Factor $4$ out of $24$ .

$f (\frac{24}{x + 12}) = 6 (4 (\frac{x + 12}{4 (6)})) - 12$

Cancel the common factor.

$f (\frac{24}{x + 12}) = 6 (4 (\frac{x + 12}{4 \cdot 6})) - 12$

Rewrite the expression.

$f (\frac{24}{x + 12}) = 6 (\frac{x + 12}{6}) - 12$

Cancel the common factor of $6$ .

Cancel the common factor.

$f (\frac{24}{x + 12}) = 6 (\frac{x + 12}{6}) - 12$

Rewrite the expression.

$f (\frac{24}{x + 12}) = x + 12 - 12$

Combine the opposite terms in $x + 12 - 12$ .

Subtract $12$ from $12$ .

$f (\frac{24}{x + 12}) = x + 0$

Add $x$ and $0$ .

$f (\frac{24}{x + 12}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{24}{x + 12}$ is the inverse of $f (x) = 6 (\frac{4}{x}) - 12$ .

$f^{- 1} (x) = \frac{24}{x + 12}$

Find the Inverse 6(4/x)-12

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