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

$f (x) = 4 x - 4$

Step 1

Write $f (x) = 4 x - 4$ as an equation.

$y = 4 x - 4$

Step 2

Interchange the variables.

$x = 4 y - 4$

Step 3

Solve for $y$ .

Rewrite the equation as $4 y - 4 = x$ .

$4 y - 4 = x$

Add $4$ to both sides of the equation.

$4 y = x + 4$

Divide each term in $4 y = x + 4$ by $4$ and simplify.

Divide each term in $4 y = x + 4$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Divide $4$ by $4$ .

$y = \frac{x}{4} + 1$

Step 4

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

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

Step 5

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

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

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

Cancel the common factor of $4 x - 4$ and $4$ .

Factor $4$ out of $4 x$ .

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

Factor $4$ out of $- 4$ .

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

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

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $x - 1$ by $1$ .

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

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

Add $- 1$ and $1$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $4$ by $1$ .

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

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

Subtract $4$ from $4$ .

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

Add $x$ and $0$ .

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

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

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

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

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