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

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

Add $2$ to both sides of the equation.

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

Multiply both sides of the equation by $4$ .

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

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$y = 4 (x + 2)$

Simplify the right side.

Simplify $4 (x + 2)$ .

Apply the distributive property.

$y = 4 x + 4 \cdot 2$

Multiply $4$ by $2$ .

$y = 4 x + 8$

Step 4

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

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

Step 5

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $4$ by $- 2$ .

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

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

Add $- 8$ and $8$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (4 x + 8)$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Cancel the common factor of $4 x + 8$ and $4$ .

Factor $4$ out of $4 x$ .

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

Factor $4$ out of $8$ .

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

Factor $4$ out of $4 (x) + 4 (2)$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $x + 2$ by $1$ .

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

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

Subtract $2$ from $2$ .

$f (4 x + 8) = x + 0$

Add $x$ and $0$ .

$f (4 x + 8) = x$

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

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

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

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