Finite Math Examples

$D^{- 1}$

Step 1

Interchange the variables.

$D = y^{- 1}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $y^{- 1} = D$ .

$y^{- 1} = D$

Step 2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{y} = D$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$y$

Step 2.4

Multiply each term in $\frac{1}{y} = D$ by $y$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{1}{y} = D$ by $y$ .

$\frac{1}{y} y = D y$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $y$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{1}{y} y = D y$

Step 2.4.2.1.2

Rewrite the expression.

$1 = D y$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $D y = 1$ .

$D y = 1$

Step 2.5.2

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

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Step 2.5.2.1

Divide each term in $D y = 1$ by $D$ .

$\frac{D y}{D} = \frac{1}{D}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $D$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{D y}{D} = \frac{1}{D}$

Step 2.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{D}$

Step 3

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Move $D^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

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

Step 4.3.3

Change the sign of the exponent by rewriting the base as its reciprocal.

$f (\frac{1}{D}) = D$

Step 4.4

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

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