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

$f (x) = 4 - 2 x$

Step 1

Write $f (x) = 4 - 2 x$ as an equation.

$y = 4 - 2 x$

Step 2

Interchange the variables.

$x = 4 - 2 y$

Step 3

Solve for $y$ .

Rewrite the equation as $4 - 2 y = x$ .

$4 - 2 y = x$

Subtract $4$ from both sides of the equation.

$- 2 y = x - 4$

Divide each term in $- 2 y = x - 4$ by $- 2$ and simplify.

Divide each term in $- 2 y = x - 4$ by $- 2$ .

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Divide $- 4$ by $- 2$ .

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

Step 4

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

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

Step 5

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

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

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

Simplify each term.

Cancel the common factor of $4 - 2 x$ and $2$ .

Factor $2$ out of $4$ .

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

Factor $2$ out of $- 2 x$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 - x$ by $1$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

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

Add $- 2$ and $2$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x}{2}$ into the numerator.

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

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 2$ by $2$ .

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

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

Subtract $4$ from $4$ .

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

Add $x$ and $0$ .

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

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

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

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

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