Algebra Examples

$- 6 x$

Step 1

Interchange the variables.

$x = - 6 y$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- 6 y = x$ .

$- 6 y = x$

Step 2.2

Divide each term in $- 6 y = x$ by $- 6$ and simplify.

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Step 2.2.1

Divide each term in $- 6 y = x$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{x}{- 6}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 6$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{x}{- 6}$

Step 2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{- 6}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$y = - \frac{x}{6}$

Step 3

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

$f^{- 1} (x) = - \frac{x}{6}$

Step 4

Verify if $f^{- 1} (x) = - \frac{x}{6}$ is the inverse of $f (x) = - 6 x$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (- 6 x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 6 x) = - \frac{- 6 x}{6}$

Step 4.2.3

Cancel the common factor of $- 6$ and $6$ .

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Step 4.2.3.1

Factor $6$ out of $- 6 x$ .

$f^{- 1} (- 6 x) = - \frac{6 (- x)}{6}$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $6$ out of $6$ .

$f^{- 1} (- 6 x) = - \frac{6 (- x)}{6 (1)}$

Step 4.2.3.2.2

Cancel the common factor.

$f^{- 1} (- 6 x) = - \frac{6 (- x)}{6 \cdot 1}$

Step 4.2.3.2.3

Rewrite the expression.

$f^{- 1} (- 6 x) = - \frac{- x}{1}$

Step 4.2.3.2.4

Divide $- x$ by $1$ .

$f^{- 1} (- 6 x) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (- \frac{x}{6})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{x}{6}) = - 6 (- \frac{x}{6})$

Step 4.3.3

Cancel the common factor of $6$ .

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Step 4.3.3.1

Move the leading negative in $- \frac{x}{6}$ into the numerator.

$f (- \frac{x}{6}) = - 6 \frac{- x}{6}$

Step 4.3.3.2

Factor $6$ out of $- 6$ .

$f (- \frac{x}{6}) = 6 (- 1) (\frac{- x}{6})$

Step 4.3.3.3

Cancel the common factor.

$f (- \frac{x}{6}) = 6 \cdot (- 1 \frac{- x}{6})$

Step 4.3.3.4

Rewrite the expression.

$f (- \frac{x}{6}) = - 1 (- x)$

Step 4.3.4

Multiply.

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Step 4.3.4.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{x}{6}) = 1 x$

Step 4.3.4.2

Multiply $x$ by $1$ .

$f (- \frac{x}{6}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x}{6}$ is the inverse of $f (x) = - 6 x$ .

$f^{- 1} (x) = - \frac{x}{6}$