Finite Math Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Add $1$ to both sides of the equation.

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

Step 3.3

Move the negative in front of the fraction.

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

Step 3.4

Find the LCD of the terms in the equation.

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Step 3.4.1

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

$y, 1, 1$

Step 3.4.2

The LCM of one and any expression is the expression.

$y$

Step 3.5

Multiply each term in $- \frac{2}{y} = x + 1$ by $y$ to eliminate the fractions.

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Step 3.5.1

Multiply each term in $- \frac{2}{y} = x + 1$ by $y$ .

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

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $y$ .

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Step 3.5.2.1.1

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

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

Step 3.5.2.1.2

Cancel the common factor.

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

Step 3.5.2.1.3

Rewrite the expression.

$- 2 = x y + 1 y$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Multiply $y$ by $1$ .

$- 2 = x y + y$

Step 3.6

Solve the equation.

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Step 3.6.1

Rewrite the equation as $x y + y = - 2$ .

$x y + y = - 2$

Step 3.6.2

Factor $y$ out of $x y + y$ .

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Step 3.6.2.1

Factor $y$ out of $x y$ .

$y x + y = - 2$

Step 3.6.2.2

Raise $y$ to the power of $1$ .

$y x + y = - 2$

Step 3.6.2.3

Factor $y$ out of $y^{1}$ .

$y x + y \cdot 1 = - 2$

Step 3.6.2.4

Factor $y$ out of $y x + y \cdot 1$ .

$y (x + 1) = - 2$

Step 3.6.3

Divide each term in $y (x + 1) = - 2$ by $x + 1$ and simplify.

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Step 3.6.3.1

Divide each term in $y (x + 1) = - 2$ by $x + 1$ .

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

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

Cancel the common factor of $x + 1$ .

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Step 3.6.3.2.1.1

Cancel the common factor.

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

Step 3.6.3.2.1.2

Divide $y$ by $1$ .

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

Step 3.6.3.3

Simplify the right side.

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Step 3.6.3.3.1

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

Add $- 1$ and $1$ .

$f^{- 1} (\frac{- 2}{x} - 1) = - \frac{2}{\frac{- 2}{x} + 0}$

Step 5.2.3.2

Add $\frac{- 2}{x}$ and $0$ .

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

Step 5.2.3.3

Move the negative in front of the fraction.

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

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.5

Cancel the common factor of $2$ .

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Step 5.2.5.1

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

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

Step 5.2.5.2

Cancel the common factor.

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

Step 5.2.5.3

Rewrite the expression.

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.3.2

Cancel the common factor of $2$ .

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Step 5.3.3.2.1

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

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

Step 5.3.3.2.2

Factor $2$ out of $- 2$ .

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

Step 5.3.3.2.3

Cancel the common factor.

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

Step 5.3.3.2.4

Rewrite the expression.

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

Step 5.3.3.3

Multiply $- 1$ by $- 1$ .

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

Step 5.3.3.4

Multiply $x + 1$ by $1$ .

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

Step 5.3.4

Combine the opposite terms in $x + 1 - 1$ .

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Step 5.3.4.1

Subtract $1$ from $1$ .

$f (- \frac{2}{x + 1}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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