Find the Inverse (x-4)/4

$\frac{x - 4}{4}$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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

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

Multiply both sides of the equation by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$y - 4 = 4 x$

Add $4$ to both sides of the equation.

$y = 4 x + 4$

Step 3

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

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

Step 4

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Add $- 4$ and $4$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Cancel the common factor of $(4 x + 4) - 4$ and $4$ .

Factor $4$ out of $4 x$ .

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

Factor $4$ out of $4$ .

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

Factor $4$ out of $4 (x) + 4 (1)$ .

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

Factor $4$ out of $- 4$ .

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

Factor $4$ out of $4 (x + 1) + 4 (- 1)$ .

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $x + 1 - 1$ by $1$ .

$f (4 x + 4) = x + 1 - 1$

Combine the opposite terms in $x + 1 - 1$ .

Subtract $1$ from $1$ .

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

Add $x$ and $0$ .

$f (4 x + 4) = x$

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

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

Find the Inverse (x-4)/4

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