Find the Inverse f(x)=1/2x-4

$f (x) = \frac{1}{2} x - 4$

Write $f (x) = \frac{1}{2} x - 4$ as an equation.

$y = \frac{1}{2} x - 4$

Interchange the variables.

$x = \frac{1}{2} y - 4$

Solve for $y$ .

Rewrite the equation as $\frac{1}{2} y - 4 = x$ .

$\frac{1}{2} y - 4 = x$

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

$\frac{y}{2} - 4 = x$

Add $4$ to both sides of the equation.

$\frac{y}{2} = x + 4$

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 (x + 4)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{y}{2} = 2 (x + 4)$

Rewrite the expression.

$y = 2 (x + 4)$

Simplify the right side.

Simplify $2 (x + 4)$ .

Apply the distributive property.

$y = 2 x + 2 \cdot 4$

Multiply $2$ by $4$ .

$y = 2 x + 8$

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

$f^{- 1} (x) = 2 x + 8$

Verify if $f^{- 1} (x) = 2 x + 8$ is the inverse of $f (x) = \frac{1}{2} x - 4$ .

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 $f^{- 1} (\frac{1}{2} x - 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{2} x - 4) = 2 (\frac{1}{2} x - 4) + 8$

Simplify each term.

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

$f^{- 1} (\frac{1}{2} x - 4) = 2 (\frac{x}{2} - 4) + 8$

Apply the distributive property.

$f^{- 1} (\frac{1}{2} x - 4) = 2 (\frac{x}{2}) + 2 \cdot - 4 + 8$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\frac{1}{2} x - 4) = 2 (\frac{x}{2}) + 2 \cdot - 4 + 8$

Rewrite the expression.

$f^{- 1} (\frac{1}{2} x - 4) = x + 2 \cdot - 4 + 8$

Multiply $2$ by $- 4$ .

$f^{- 1} (\frac{1}{2} x - 4) = x - 8 + 8$

Combine the opposite terms in $x - 8 + 8$ .

Add $- 8$ and $8$ .

$f^{- 1} (\frac{1}{2} x - 4) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{1}{2} x - 4) = x$

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

Set up the composite result function.

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

Evaluate $f (2 x + 8)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (2 x + 8) = \frac{1}{2} \cdot (2 x + 8) - 4$

Simplify each term.

Apply the distributive property.

$f (2 x + 8) = \frac{1}{2} \cdot (2 x) + \frac{1}{2} \cdot 8 - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

$f (2 x + 8) = \frac{1}{2} \cdot (2 (x)) + \frac{1}{2} \cdot 8 - 4$

Cancel the common factor.

$f (2 x + 8) = \frac{1}{2} \cdot (2 x) + \frac{1}{2} \cdot 8 - 4$

Rewrite the expression.

$f (2 x + 8) = x + \frac{1}{2} \cdot 8 - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$f (2 x + 8) = x + \frac{1}{2} \cdot (2 (4)) - 4$

Cancel the common factor.

$f (2 x + 8) = x + \frac{1}{2} \cdot (2 \cdot 4) - 4$

Rewrite the expression.

$f (2 x + 8) = x + 4 - 4$

Combine the opposite terms in $x + 4 - 4$ .

Subtract $4$ from $4$ .

$f (2 x + 8) = x + 0$

Add $x$ and $0$ .

$f (2 x + 8) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 2 x + 8$ is the inverse of $f (x) = \frac{1}{2} x - 4$ .

$f^{- 1} (x) = 2 x + 8$

Find the Inverse f(x)=1/2x-4

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