Find the Inverse f(x)=4x-8

$f (x) = 4 x - 8$

Step 1

Write $f (x) = 4 x - 8$ as an equation.

$y = 4 x - 8$

Step 2

Interchange the variables.

$x = 4 y - 8$

Step 3

Solve for $y$ .

Rewrite the equation as $4 y - 8 = x$ .

$4 y - 8 = x$

Add $8$ to both sides of the equation.

$4 y = x + 8$

Divide each term in $4 y = x + 8$ by $4$ and simplify.

Divide each term in $4 y = x + 8$ by $4$ .

$\frac{4 y}{4} = \frac{x}{4} + \frac{8}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{x}{4} + \frac{8}{4}$

Divide $y$ by $1$ .

$y = \frac{x}{4} + \frac{8}{4}$

Simplify the right side.

Divide $8$ by $4$ .

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

Step 4

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

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

Step 5

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

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

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

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

Factor $4$ out of $4 x$ .

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

Factor $4$ out of $- 8$ .

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

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

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $x - 2$ by $1$ .

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

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

Add $- 2$ and $2$ .

$f^{- 1} (4 x - 8) = x + 0$

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $4$ by $2$ .

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

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

Subtract $8$ from $8$ .

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

Add $x$ and $0$ .

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

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

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

Find the Inverse f(x)=4x-8

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