Find the Inverse 3X-2

$3 X - 2$

Step 1

Interchange the variables.

$X = 3 y - 2$

Step 2

Solve for $y$ .

Rewrite the equation as $3 y - 2 = X$ .

$3 y - 2 = X$

Add $2$ to both sides of the equation.

$3 y = X + 2$

Divide each term in $3 y = X + 2$ by $3$ and simplify.

Divide each term in $3 y = X + 2$ by $3$ .

$\frac{3 y}{3} = \frac{X}{3} + \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{X}{3} + \frac{2}{3}$

Divide $y$ by $1$ .

$y = \frac{X}{3} + \frac{2}{3}$

Step 3

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

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

Step 4

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

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

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

Combine the numerators over the common denominator.

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

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

Add $- 2$ and $2$ .

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

Add $3 X$ and $0$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $X$ by $1$ .

$f^{- 1} (3 X - 2) = X$

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Subtract $2$ from $2$ .

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

Add $X$ and $0$ .

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

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

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

Find the Inverse 3X-2

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