Find the Inverse f(x)=6x-6

$f (x) = 6 x - 6$

Step 1

Write $f (x) = 6 x - 6$ as an equation.

$y = 6 x - 6$

Step 2

Interchange the variables.

$x = 6 y - 6$

Step 3

Solve for $y$ .

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

$6 y - 6 = x$

Add $6$ to both sides of the equation.

$6 y = x + 6$

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Divide $6$ by $6$ .

$y = \frac{x}{6} + 1$

Step 4

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

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

Step 5

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

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

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

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

Factor $6$ out of $6 x$ .

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

Factor $6$ out of $- 6$ .

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

Factor $6$ out of $6 (x) + 6 (- 1)$ .

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

Cancel the common factors.

Factor $6$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $x - 1$ by $1$ .

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

Combine the opposite terms in $x - 1 + 1$ .

Add $- 1$ and $1$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $6$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $6$ by $1$ .

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

Combine the opposite terms in $x + 6 - 6$ .

Subtract $6$ from $6$ .

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

Add $x$ and $0$ .

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

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

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

Find the Inverse f(x)=6x-6

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