Find the Inverse -12x+6

$- 12 x + 6$

Step 1

Interchange the variables.

$x = - 12 y + 6$

Step 2

Solve for $y$ .

Rewrite the equation as $- 12 y + 6 = x$ .

$- 12 y + 6 = x$

Subtract $6$ from both sides of the equation.

$- 12 y = x - 6$

Divide each term in $- 12 y = x - 6$ by $- 12$ and simplify.

Divide each term in $- 12 y = x - 6$ by $- 12$ .

$\frac{- 12 y}{- 12} = \frac{x}{- 12} + \frac{- 6}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 y}{- 12} = \frac{x}{- 12} + \frac{- 6}{- 12}$

Divide $y$ by $1$ .

$y = \frac{x}{- 12} + \frac{- 6}{- 12}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{x}{12} + \frac{- 6}{- 12}$

Cancel the common factor of $- 6$ and $- 12$ .

Factor $- 6$ out of $- 6$ .

$y = - \frac{x}{12} + \frac{- 6 (1)}{- 12}$

Cancel the common factors.

Factor $- 6$ out of $- 12$ .

$y = - \frac{x}{12} + \frac{- 6 \cdot 1}{- 6 \cdot 2}$

Cancel the common factor.

$y = - \frac{x}{12} + \frac{- 6 \cdot 1}{- 6 \cdot 2}$

Rewrite the expression.

$y = - \frac{x}{12} + \frac{1}{2}$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = - \frac{x}{12} + \frac{1}{2}$ is the inverse of $f (x) = - 12 x + 6$ .

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

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

Simplify terms.

Cancel the common factor of $- 12 x + 6$ and $12$ .

Factor $6$ out of $- 12 x$ .

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

Factor $6$ out of $6$ .

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

Factor $6$ out of $6 (- 2 x) + 6 (1)$ .

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

Cancel the common factors.

Factor $6$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Multiply $- 2$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

Simplify terms.

Combine the opposite terms in $2 x - 1 + 1$ .

Add $- 1$ and $1$ .

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

Add $2 x$ and $0$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

$f^{- 1} (- 12 x + 6) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{x}{12} + \frac{1}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $12$ .

Move the leading negative in $- \frac{x}{12}$ into the numerator.

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

Factor $12$ out of $- 12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Cancel the common factor of $2$ .

Factor $2$ out of $- 12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Add $- 6$ and $6$ .

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

Add $x$ and $0$ .

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

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x}{12} + \frac{1}{2}$ is the inverse of $f (x) = - 12 x + 6$ .

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

Find the Inverse -12x+6

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