Find the Inverse x/6+9

$\frac{x}{6} + 9$

Step 1

Interchange the variables.

$x = \frac{y}{6} + 9$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y}{6} + 9 = x$ .

$\frac{y}{6} + 9 = x$

Subtract $9$ from both sides of the equation.

$\frac{y}{6} = x - 9$

Multiply both sides of the equation by $6$ .

$6 \frac{y}{6} = 6 (x - 9)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$6 \frac{y}{6} = 6 (x - 9)$

Rewrite the expression.

$y = 6 (x - 9)$

Simplify the right side.

Simplify $6 (x - 9)$ .

Apply the distributive property.

$y = 6 x + 6 \cdot - 9$

Multiply $6$ by $- 9$ .

$y = 6 x - 54$

Step 3

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

$f^{- 1} (x) = 6 x - 54$

Step 4

Verify if $f^{- 1} (x) = 6 x - 54$ is the inverse of $f (x) = \frac{x}{6} + 9$ .

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 $f^{- 1} (\frac{x}{6} + 9)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x}{6} + 9) = 6 (\frac{x}{6} + 9) - 54$

Simplify each term.

Apply the distributive property.

$f^{- 1} (\frac{x}{6} + 9) = 6 (\frac{x}{6}) + 6 \cdot 9 - 54$

Cancel the common factor of $6$ .

Cancel the common factor.

$f^{- 1} (\frac{x}{6} + 9) = 6 (\frac{x}{6}) + 6 \cdot 9 - 54$

Rewrite the expression.

$f^{- 1} (\frac{x}{6} + 9) = x + 6 \cdot 9 - 54$

Multiply $6$ by $9$ .

$f^{- 1} (\frac{x}{6} + 9) = x + 54 - 54$

Combine the opposite terms in $x + 54 - 54$ .

Subtract $54$ from $54$ .

$f^{- 1} (\frac{x}{6} + 9) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x}{6} + 9) = x$

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

Set up the composite result function.

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

Evaluate $f (6 x - 54)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (6 x - 54) = \frac{6 x - 54}{6} + 9$

Cancel the common factor of $6 x - 54$ and $6$ .

Factor $6$ out of $6 x$ .

$f (6 x - 54) = \frac{6 (x) - 54}{6} + 9$

Factor $6$ out of $- 54$ .

$f (6 x - 54) = \frac{6 (x) + 6 \cdot - 9}{6} + 9$

Factor $6$ out of $6 (x) + 6 (- 9)$ .

$f (6 x - 54) = \frac{6 (x - 9)}{6} + 9$

Cancel the common factors.

Factor $6$ out of $6$ .

$f (6 x - 54) = \frac{6 (x - 9)}{6 (1)} + 9$

Cancel the common factor.

$f (6 x - 54) = \frac{6 (x - 9)}{6 \cdot 1} + 9$

Rewrite the expression.

$f (6 x - 54) = \frac{x - 9}{1} + 9$

Divide $x - 9$ by $1$ .

$f (6 x - 54) = x - 9 + 9$

Combine the opposite terms in $x - 9 + 9$ .

Add $- 9$ and $9$ .

$f (6 x - 54) = x + 0$

Add $x$ and $0$ .

$f (6 x - 54) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 6 x - 54$ is the inverse of $f (x) = \frac{x}{6} + 9$ .

$f^{- 1} (x) = 6 x - 54$

Find the Inverse x/6+9

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