Precalculus Examples

$f (x) = - 1 x$

Step 1

Write $f (x) = - 1 x$ as an equation.

$y = - 1 x$

Step 2

Interchange the variables.

$x = - 1 y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 1 y = x$ .

$- 1 y = x$

Step 3.2

Divide each term in $- 1 y = x$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- 1 y = x$ by $- 1$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2.2

Divide $y$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$y = - 1 \cdot x$

Step 3.2.3.2

Rewrite $- 1 \cdot x$ as $- x$ .

$y = - x$

Step 4

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

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

Step 5

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

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Rewrite $- 1 x$ as $- x$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

Multiply $- 1 (- x)$ .

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Step 5.3.3.1

Multiply $- 1$ by $- 1$ .

$f (- x) = 1 x$

Step 5.3.3.2

Multiply $x$ by $1$ .

$f (- x) = x$

Step 5.4

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

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