Find the Inverse f(x)=6x-2

$f (x) = 6 x - 2$

Step 1

Write $f (x) = 6 x - 2$ as an equation.

$y = 6 x - 2$

Step 2

Interchange the variables.

$x = 6 y - 2$

Step 3

Solve for $y$ .

Rewrite the equation as $6 y - 2 = x$ .

$6 y - 2 = x$

Add $2$ to both sides of the equation.

$6 y = x + 2$

Divide each term in $6 y = x + 2$ by $6$ and simplify.

Divide each term in $6 y = x + 2$ by $6$ .

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2$ .

$y = \frac{x}{6} + \frac{2 (1)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$y = \frac{x}{6} + \frac{2 \cdot 1}{2 \cdot 3}$

Cancel the common factor.

$y = \frac{x}{6} + \frac{2 \cdot 1}{2 \cdot 3}$

Rewrite the expression.

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

Step 4

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

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

Step 5

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

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

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

Cancel the common factor of $6 x - 2$ and $6$ .

Factor $2$ out of $6 x$ .

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

Factor $2$ out of $- 2$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

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

Add $- 1$ and $1$ .

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

Add $3 x$ and $0$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $6$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Subtract $2$ from $2$ .

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

Add $x$ and $0$ .

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

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

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

Find the Inverse f(x)=6x-2

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