Algebra Examples

$f (x) = - \frac{x}{12}$

Step 1

Write $f (x) = - \frac{x}{12}$ as an equation.

$y = - \frac{x}{12}$

Step 2

Interchange the variables.

$x = - \frac{y}{12}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \frac{y}{12} = x$ .

$- \frac{y}{12} = x$

Step 3.2

Multiply both sides of the equation by $- 12$ .

$- 12 (- \frac{y}{12}) = - 12 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify $- 12 (- \frac{y}{12})$ .

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Step 3.3.1.1

Cancel the common factor of $12$ .

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Step 3.3.1.1.1

Move the leading negative in $- \frac{y}{12}$ into the numerator.

$- 12 \frac{- y}{12} = - 12 x$

Step 3.3.1.1.2

Factor $12$ out of $- 12$ .

$12 (- 1) \frac{- y}{12} = - 12 x$

Step 3.3.1.1.3

Cancel the common factor.

$12 \cdot - 1 \frac{- y}{12} = - 12 x$

Step 3.3.1.1.4

Rewrite the expression.

$- - y = - 12 x$

Step 3.3.1.2

Multiply.

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Step 3.3.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - 12 x$

Step 3.3.1.2.2

Multiply $y$ by $1$ .

$y = - 12 x$

Step 4

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

$f^{- 1} (x) = - 12 x$

Step 5

Verify if $f^{- 1} (x) = - 12 x$ is the inverse of $f (x) = - \frac{x}{12}$ .

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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 $f^{- 1} (- \frac{x}{12})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{x}{12}) = - 12 (- \frac{x}{12})$

Step 5.2.3

Cancel the common factor of $12$ .

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Step 5.2.3.1

Move the leading negative in $- \frac{x}{12}$ into the numerator.

$f^{- 1} (- \frac{x}{12}) = - 12 \frac{- x}{12}$

Step 5.2.3.2

Factor $12$ out of $- 12$ .

$f^{- 1} (- \frac{x}{12}) = 12 (- 1) (\frac{- x}{12})$

Step 5.2.3.3

Cancel the common factor.

$f^{- 1} (- \frac{x}{12}) = 12 \cdot (- 1 \frac{- x}{12})$

Step 5.2.3.4

Rewrite the expression.

$f^{- 1} (- \frac{x}{12}) = - 1 (- x)$

Step 5.2.4

Multiply.

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Step 5.2.4.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \frac{x}{12}) = 1 x$

Step 5.2.4.2

Multiply $x$ by $1$ .

$f^{- 1} (- \frac{x}{12}) = x$

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 $f (- 12 x)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- 12 x) = - \frac{- 12 x}{12}$

Step 5.3.3

Cancel the common factor of $- 12$ and $12$ .

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Step 5.3.3.1

Factor $12$ out of $- 12 x$ .

$f (- 12 x) = - \frac{12 (- x)}{12}$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $12$ out of $12$ .

$f (- 12 x) = - \frac{12 (- x)}{12 (1)}$

Step 5.3.3.2.2

Cancel the common factor.

$f (- 12 x) = - \frac{12 (- x)}{12 \cdot 1}$

Step 5.3.3.2.3

Rewrite the expression.

$f (- 12 x) = - \frac{- x}{1}$

Step 5.3.3.2.4

Divide $- x$ by $1$ .

$f (- 12 x) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - 12 x$ is the inverse of $f (x) = - \frac{x}{12}$ .

$f^{- 1} (x) = - 12 x$