Find the Inverse f(x)=(3x+1)/6

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

Step 1

Write $f (x) = \frac{3 x + 1}{6}$ as an equation.

$y = \frac{3 x + 1}{6}$

Step 2

Interchange the variables.

$x = \frac{3 y + 1}{6}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{3 y + 1}{6} = x$ .

$\frac{3 y + 1}{6} = x$

Multiply both sides by $6$ .

$\frac{3 y + 1}{6} \cdot 6 = x \cdot 6$

Simplify.

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{3 y + 1}{6} \cdot 6 = x \cdot 6$

Rewrite the expression.

$3 y + 1 = x \cdot 6$

Simplify the right side.

Move $6$ to the left of $x$ .

$3 y + 1 = 6 x$

Solve for $y$ .

Subtract $1$ from both sides of the equation.

$3 y = 6 x - 1$

Divide each term in $3 y = 6 x - 1$ by $3$ and simplify.

Divide each term in $3 y = 6 x - 1$ by $3$ .

$\frac{3 y}{3} = \frac{6 x}{3} + \frac{- 1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{6 x}{3} + \frac{- 1}{3}$

Divide $y$ by $1$ .

$y = \frac{6 x}{3} + \frac{- 1}{3}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 x$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

$y = \frac{2 x}{1} + \frac{- 1}{3}$

Divide $2 x$ by $1$ .

$y = 2 x + \frac{- 1}{3}$

Move the negative in front of the fraction.

$y = 2 x - \frac{1}{3}$

Step 4

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

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

Step 5

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

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Combine the opposite terms in $3 x + 1 - 1$ .

Subtract $1$ from $1$ .

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

Add $3 x$ and $0$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $3$ .

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Add $- 1$ and $1$ .

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

Add $6 x$ and $0$ .

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

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

Find the Inverse f(x)=(3x+1)/6

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