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

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

Step 1

Write $f (x) = \frac{1}{2} x + \frac{3}{2}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

Combine $\frac{1}{2}$ and $y$ .

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

Subtract $\frac{3}{2}$ from both sides of the equation.

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

Multiply both sides of the equation by $2$ .

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

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

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

Apply the distributive property.

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

$y = 2 x - 3$

Step 4

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

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

Step 5

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

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

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

Simplify each term.

Combine $\frac{1}{2}$ and $x$ .

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

Apply the distributive property.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Subtract $3$ from $3$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Move the negative in front of the fraction.

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

Combine the opposite terms in $x - \frac{3}{2} + \frac{3}{2}$ .

Add $- \frac{3}{2}$ and $\frac{3}{2}$ .

$f (2 x - 3) = x + 0$

Add $x$ and $0$ .

$f (2 x - 3) = x$

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

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

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

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