Find the Inverse 6(x-2)+3

$6 (x - 2) + 3$

Step 1

Interchange the variables.

$x = 6 (y - 2) + 3$

Step 2

Solve for $y$ .

Rewrite the equation as $6 (y - 2) + 3 = x$ .

$6 (y - 2) + 3 = x$

Simplify $6 (y - 2) + 3$ .

Simplify each term.

Apply the distributive property.

$6 y + 6 \cdot - 2 + 3 = x$

Multiply $6$ by $- 2$ .

$6 y - 12 + 3 = x$

Add $- 12$ and $3$ .

$6 y - 9 = x$

Add $9$ to both sides of the equation.

$6 y = x + 9$

Divide each term in $6 y = x + 9$ by $6$ and simplify.

Divide each term in $6 y = x + 9$ by $6$ .

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $9$ and $6$ .

Factor $3$ out of $9$ .

$y = \frac{x}{6} + \frac{3 (3)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

$y = \frac{x}{6} + \frac{3 \cdot 3}{3 \cdot 2}$

Cancel the common factor.

$y = \frac{x}{6} + \frac{3 \cdot 3}{3 \cdot 2}$

Rewrite the expression.

$y = \frac{x}{6} + \frac{3}{2}$

Step 3

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

$f^{- 1} (x) = \frac{x}{6} + \frac{3}{2}$

Step 4

Verify if $f^{- 1} (x) = \frac{x}{6} + \frac{3}{2}$ is the inverse of $f (x) = 6 (x - 2) + 3$ .

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 $f^{- 1} (6 (x - 2) + 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (6 (x - 2) + 3) = \frac{6 (x - 2) + 3}{6} + \frac{3}{2}$

Simplify each term.

Cancel the common factor of $6 (x - 2) + 3$ and $6$ .

Factor $3$ out of $6 (x - 2)$ .

$f^{- 1} (6 (x - 2) + 3) = \frac{3 (2 (x - 2)) + 3}{6} + \frac{3}{2}$

Factor $3$ out of $3$ .

$f^{- 1} (6 (x - 2) + 3) = \frac{3 (2 (x - 2)) + 3 (1)}{6} + \frac{3}{2}$

Factor $3$ out of $3 (2 (x - 2)) + 3 (1)$ .

$f^{- 1} (6 (x - 2) + 3) = \frac{3 (2 (x - 2) + 1)}{6} + \frac{3}{2}$

Cancel the common factors.

Factor $3$ out of $6$ .

$f^{- 1} (6 (x - 2) + 3) = \frac{3 (2 (x - 2) + 1)}{3 (2)} + \frac{3}{2}$

Cancel the common factor.

$f^{- 1} (6 (x - 2) + 3) = \frac{3 (2 (x - 2) + 1)}{3 \cdot 2} + \frac{3}{2}$

Rewrite the expression.

$f^{- 1} (6 (x - 2) + 3) = \frac{2 (x - 2) + 1}{2} + \frac{3}{2}$

Simplify the numerator.

Apply the distributive property.

$f^{- 1} (6 (x - 2) + 3) = \frac{2 x + 2 \cdot - 2 + 1}{2} + \frac{3}{2}$

Multiply $2$ by $- 2$ .

$f^{- 1} (6 (x - 2) + 3) = \frac{2 x - 4 + 1}{2} + \frac{3}{2}$

Add $- 4$ and $1$ .

$f^{- 1} (6 (x - 2) + 3) = \frac{2 x - 3}{2} + \frac{3}{2}$

Simplify terms.

Combine the numerators over the common denominator.

$f^{- 1} (6 (x - 2) + 3) = \frac{2 x - 3 + 3}{2}$

Combine the opposite terms in $2 x - 3 + 3$ .

Add $- 3$ and $3$ .

$f^{- 1} (6 (x - 2) + 3) = \frac{2 x + 0}{2}$

Add $2 x$ and $0$ .

$f^{- 1} (6 (x - 2) + 3) = \frac{2 x}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (6 (x - 2) + 3) = \frac{2 x}{2}$

Divide $x$ by $1$ .

$f^{- 1} (6 (x - 2) + 3) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{6} + \frac{3}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{6} + \frac{3}{2}) = 6 ((\frac{x}{6} + \frac{3}{2}) - 2) + 3$

Simplify each term.

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{x}{6} + \frac{3}{2}) = 6 (\frac{x}{6} + \frac{3}{2} - 2 \cdot \frac{2}{2}) + 3$

Combine $- 2$ and $\frac{2}{2}$ .

$f (\frac{x}{6} + \frac{3}{2}) = 6 (\frac{x}{6} + \frac{3}{2} + \frac{- 2 \cdot 2}{2}) + 3$

Combine the numerators over the common denominator.

$f (\frac{x}{6} + \frac{3}{2}) = 6 (\frac{x}{6} + \frac{3 - 2 \cdot 2}{2}) + 3$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$f (\frac{x}{6} + \frac{3}{2}) = 6 (\frac{x}{6} + \frac{3 - 4}{2}) + 3$

Subtract $4$ from $3$ .

$f (\frac{x}{6} + \frac{3}{2}) = 6 (\frac{x}{6} + \frac{- 1}{2}) + 3$

Move the negative in front of the fraction.

$f (\frac{x}{6} + \frac{3}{2}) = 6 (\frac{x}{6} - \frac{1}{2}) + 3$

Apply the distributive property.

$f (\frac{x}{6} + \frac{3}{2}) = 6 (\frac{x}{6}) + 6 (- \frac{1}{2}) + 3$

Cancel the common factor of $6$ .

Cancel the common factor.

$f (\frac{x}{6} + \frac{3}{2}) = 6 (\frac{x}{6}) + 6 (- \frac{1}{2}) + 3$

Rewrite the expression.

$f (\frac{x}{6} + \frac{3}{2}) = x + 6 (- \frac{1}{2}) + 3$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (\frac{x}{6} + \frac{3}{2}) = x + 6 (\frac{- 1}{2}) + 3$

Factor $2$ out of $6$ .

$f (\frac{x}{6} + \frac{3}{2}) = x + 2 (3) (\frac{- 1}{2}) + 3$

Cancel the common factor.

$f (\frac{x}{6} + \frac{3}{2}) = x + 2 \cdot (3 (\frac{- 1}{2})) + 3$

Rewrite the expression.

$f (\frac{x}{6} + \frac{3}{2}) = x + 3 \cdot - 1 + 3$

Multiply $3$ by $- 1$ .

$f (\frac{x}{6} + \frac{3}{2}) = x - 3 + 3$

Combine the opposite terms in $x - 3 + 3$ .

Add $- 3$ and $3$ .

$f (\frac{x}{6} + \frac{3}{2}) = x + 0$

Add $x$ and $0$ .

$f (\frac{x}{6} + \frac{3}{2}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{6} + \frac{3}{2}$ is the inverse of $f (x) = 6 (x - 2) + 3$ .

$f^{- 1} (x) = \frac{x}{6} + \frac{3}{2}$

Find the Inverse 6(x-2)+3

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