Find the Inverse Function f(x)=3/2x+1

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

Replace $f (x)$ with $y$ .

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

Interchange the variables.

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

Solve for $y$ .

Tap for more steps…

Rewrite the equation as $\frac{3}{2} y + 1 = x$ .

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

Combine $\frac{3}{2}$ and $y$ .

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

Subtract $1$ from both sides of the equation.

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

Multiply both sides of the equation by $\frac{2}{3}$ .

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

Simplify both sides of the equation.

Tap for more steps…

Simplify $\frac{2}{3} \cdot \frac{3 y}{2}$ .

Tap for more steps…

Cancel the common factor of $2$ .

Tap for more steps…

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Tap for more steps…

Factor $3$ out of $3 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify $\frac{2}{3} \cdot (x - 1)$ .

Tap for more steps…

Apply the distributive property.

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

Combine $\frac{2}{3}$ and $x$ .

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

Multiply $\frac{2}{3} \cdot - 1$ .

Tap for more steps…

Combine $\frac{2}{3}$ and $- 1$ .

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

Multiply $2$ by $- 1$ .

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

Move the negative in front of the fraction.

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

Solve for $y$ and replace with $f^{- 1} (x)$ .

Tap for more steps…

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

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

Set up the composite result function.

$g (f (x))$

Evaluate $g (f (x))$ by substituting in the value of $f$ into $g$ .

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

Simplify terms.

Tap for more steps…

Simplify each term.

Tap for more steps…

Simplify the numerator.

Tap for more steps…

Combine $\frac{3}{2}$ and $x$ .

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Combine $2$ and $\frac{3 x + 2}{2}$ .

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

Reduce the expression by cancelling the common factors.

Tap for more steps…

Reduce the expression $\frac{2 (3 x + 2)}{2}$ by cancelling the common factors.

Tap for more steps…

Cancel the common factor.

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

Rewrite the expression.

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

Divide $3 x + 2$ by $1$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Tap for more steps…

Subtract $2$ from $2$ .

$g (\frac{3}{2} x + 1) = \frac{3 x + 0}{3}$

Add $3 x$ and $0$ .

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

Cancel the common factor of $3$ .

Tap for more steps…

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

Find the Inverse Function f(x)=3/2x+1

Need help with MATH HOMEWORK