Find the Inverse f(x)=-1/2x

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

Step 1

Write $f (x) = - \frac{1}{2} x$ as an equation.

$y = - \frac{1}{2} x$

Step 2

Interchange the variables.

$x = - \frac{1}{2} y$

Step 3

Solve for $y$ .

Rewrite the equation as $- \frac{1}{2} y = x$ .

$- \frac{1}{2} y = x$

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{1}{2} y) = - 2 x$

Simplify the left side.

Simplify $- 2 (- \frac{1}{2} y)$ .

Combine $y$ and $\frac{1}{2}$ .

$- 2 (- \frac{y}{2}) = - 2 x$

Cancel the common factor of $2$ .

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

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

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- y}{2} = - 2 x$

Cancel the common factor.

$2 \cdot - 1 \frac{- y}{2} = - 2 x$

Rewrite the expression.

$- - y = - 2 x$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - 2 x$

Multiply $y$ by $1$ .

$y = - 2 x$

Step 4

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

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

Step 5

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

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

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

Combine $x$ and $\frac{1}{2}$ .

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

Cancel the common factor of $2$ .

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

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

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

$f (- 2 x) = - 1 \cdot (- x)$

Multiply.

Multiply $- 1$ by $- 1$ .

$f (- 2 x) = 1 \cdot x$

Multiply $x$ by $1$ .

$f (- 2 x) = x$

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

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

Find the Inverse f(x)=-1/2x

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