Finite Math Examples

$y = \frac{1}{- 8 x + 1}$

Step 1

Interchange the variables.

$x = \frac{1}{- 8 y + 1}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1}{- 8 y + 1} = x$ .

$\frac{1}{- 8 y + 1} = x$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$- 8 y + 1, 1$

Step 2.2.2

Remove parentheses.

$- 8 y + 1, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$- 8 y + 1$

Step 2.3

Multiply each term in $\frac{1}{- 8 y + 1} = x$ by $- 8 y + 1$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{1}{- 8 y + 1} = x$ by $- 8 y + 1$ .

$\frac{1}{- 8 y + 1} (- 8 y + 1) = x (- 8 y + 1)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Multiply $\frac{1}{- 8 y + 1}$ by $- 8 y + 1$ .

$\frac{- 8 y + 1}{- 8 y + 1} = x (- 8 y + 1)$

Step 2.3.2.2

Cancel the common factor of $- 8 y + 1$ .

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Step 2.3.2.2.1

Cancel the common factor.

$\frac{- 8 y + 1}{- 8 y + 1} = x (- 8 y + 1)$

Step 2.3.2.2.2

Rewrite the expression.

$1 = x (- 8 y + 1)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$1 = x (- 8 y) + x \cdot 1$

Step 2.3.3.2

Simplify the expression.

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Step 2.3.3.2.1

Rewrite using the commutative property of multiplication.

$1 = - 8 x y + x \cdot 1$

Step 2.3.3.2.2

Multiply $x$ by $1$ .

$1 = - 8 x y + x$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $- 8 x y + x = 1$ .

$- 8 x y + x = 1$

Step 2.4.2

Subtract $x$ from both sides of the equation.

$- 8 x y = 1 - x$

Step 2.4.3

Divide each term in $- 8 x y = 1 - x$ by $- 8 x$ and simplify.

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Step 2.4.3.1

Divide each term in $- 8 x y = 1 - x$ by $- 8 x$ .

$\frac{- 8 x y}{- 8 x} = \frac{1}{- 8 x} + \frac{- x}{- 8 x}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $- 8$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{- 8 x y}{- 8 x} = \frac{1}{- 8 x} + \frac{- x}{- 8 x}$

Step 2.4.3.2.1.2

Rewrite the expression.

$\frac{x y}{x} = \frac{1}{- 8 x} + \frac{- x}{- 8 x}$

Step 2.4.3.2.2

Cancel the common factor of $x$ .

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Step 2.4.3.2.2.1

Cancel the common factor.

$\frac{x y}{x} = \frac{1}{- 8 x} + \frac{- x}{- 8 x}$

Step 2.4.3.2.2.2

Divide $y$ by $1$ .

$y = \frac{1}{- 8 x} + \frac{- x}{- 8 x}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Simplify each term.

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Step 2.4.3.3.1.1

Move the negative in front of the fraction.

$y = - \frac{1}{8 x} + \frac{- x}{- 8 x}$

Step 2.4.3.3.1.2

Cancel the common factor of $x$ .

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Step 2.4.3.3.1.2.1

Cancel the common factor.

$y = - \frac{1}{8 x} + \frac{- x}{- 8 x}$

Step 2.4.3.3.1.2.2

Rewrite the expression.

$y = - \frac{1}{8 x} + \frac{- 1}{- 8}$

Step 2.4.3.3.1.3

Dividing two negative values results in a positive value.

$y = - \frac{1}{8 x} + \frac{1}{8}$

Step 3

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

$f^{- 1} (x) = - \frac{1}{8 x} + \frac{1}{8}$

Step 4

Verify if $f^{- 1} (x) = - \frac{1}{8 x} + \frac{1}{8}$ is the inverse of $f (x) = \frac{1}{- 8 x + 1}$ .

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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 $f^{- 1} (\frac{1}{- 8 x + 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{- 8 x + 1}) = - \frac{1}{8 (\frac{1}{- 8 x + 1})} + \frac{1}{8}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Combine $8$ and $\frac{1}{- 8 x + 1}$ .

$f^{- 1} (\frac{1}{- 8 x + 1}) = - \frac{1}{\frac{8}{- 8 x + 1}} + \frac{1}{8}$

Step 4.2.3.2

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{1}{- 8 x + 1}) = - (1 (\frac{- 8 x + 1}{8})) + \frac{1}{8}$

Step 4.2.3.3

Multiply $\frac{- 8 x + 1}{8}$ by $1$ .

$f^{- 1} (\frac{1}{- 8 x + 1}) = - \frac{- 8 x + 1}{8} + \frac{1}{8}$

Step 4.2.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{1}{- 8 x + 1}) = \frac{- (- 8 x + 1) + 1}{8}$

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

Apply the distributive property.

$f^{- 1} (\frac{1}{- 8 x + 1}) = \frac{- (- 8 x) - 1 \cdot 1 + 1}{8}$

Step 4.2.5.2

Multiply $- 8$ by $- 1$ .

$f^{- 1} (\frac{1}{- 8 x + 1}) = \frac{8 x - 1 \cdot 1 + 1}{8}$

Step 4.2.5.3

Multiply $- 1$ by $1$ .

$f^{- 1} (\frac{1}{- 8 x + 1}) = \frac{8 x - 1 + 1}{8}$

Step 4.2.6

Simplify terms.

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Step 4.2.6.1

Combine the opposite terms in $8 x - 1 + 1$ .

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Step 4.2.6.1.1

Add $- 1$ and $1$ .

$f^{- 1} (\frac{1}{- 8 x + 1}) = \frac{8 x + 0}{8}$

Step 4.2.6.1.2

Add $8 x$ and $0$ .

$f^{- 1} (\frac{1}{- 8 x + 1}) = \frac{8 x}{8}$

Step 4.2.6.2

Cancel the common factor of $8$ .

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Step 4.2.6.2.1

Cancel the common factor.

$f^{- 1} (\frac{1}{- 8 x + 1}) = \frac{8 x}{8}$

Step 4.2.6.2.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{1}{- 8 x + 1}) = x$

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 $f (- \frac{1}{8 x} + \frac{1}{8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{- 8 (- \frac{1}{8 x} + \frac{1}{8}) + 1}$

Step 4.3.3

Simplify the denominator.

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Step 4.3.3.1

Apply the distributive property.

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{- 8 (- \frac{1}{8 x}) - 8 (\frac{1}{8}) + 1}$

Step 4.3.3.2

Cancel the common factor of $8$ .

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Step 4.3.3.2.1

Move the leading negative in $- \frac{1}{8 x}$ into the numerator.

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{- 8 \frac{- 1}{8 x} - 8 (\frac{1}{8}) + 1}$

Step 4.3.3.2.2

Factor $8$ out of $- 8$ .

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{8 (- 1) (\frac{- 1}{8 x}) - 8 (\frac{1}{8}) + 1}$

Step 4.3.3.2.3

Factor $8$ out of $8 x$ .

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{8 (- 1) (\frac{- 1}{8 (x)}) - 8 (\frac{1}{8}) + 1}$

Step 4.3.3.2.4

Cancel the common factor.

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{8 \cdot (- 1 \frac{- 1}{8 x}) - 8 (\frac{1}{8}) + 1}$

Step 4.3.3.2.5

Rewrite the expression.

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{- 1 \frac{- 1}{x} - 8 (\frac{1}{8}) + 1}$

Step 4.3.3.3

Cancel the common factor of $8$ .

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Step 4.3.3.3.1

Factor $8$ out of $- 8$ .

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{- 1 \frac{- 1}{x} + 8 (- 1) (\frac{1}{8}) + 1}$

Step 4.3.3.3.2

Cancel the common factor.

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{- 1 \frac{- 1}{x} + 8 \cdot (- 1 (\frac{1}{8})) + 1}$

Step 4.3.3.3.3

Rewrite the expression.

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{- 1 \frac{- 1}{x} - 1 + 1}$

Step 4.3.3.4

Simplify each term.

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Step 4.3.3.4.1

Move the negative in front of the fraction.

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{- 1 (- \frac{1}{x}) - 1 + 1}$

Step 4.3.3.4.2

Multiply $- 1 (- \frac{1}{x})$ .

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Step 4.3.3.4.2.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{1 (\frac{1}{x}) - 1 + 1}$

Step 4.3.3.4.2.2

Multiply $\frac{1}{x}$ by $1$ .

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{\frac{1}{x} - 1 + 1}$

Step 4.3.3.5

Add $- 1$ and $1$ .

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{\frac{1}{x} + 0}$

Step 4.3.3.6

Add $\frac{1}{x}$ and $0$ .

$f (- \frac{1}{8 x} + \frac{1}{8}) = \frac{1}{\frac{1}{x}}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{1}{8 x} + \frac{1}{8}) = 1 x$

Step 4.3.5

Multiply $x$ by $1$ .

$f (- \frac{1}{8 x} + \frac{1}{8}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{1}{8 x} + \frac{1}{8}$ is the inverse of $f (x) = \frac{1}{- 8 x + 1}$ .

$f^{- 1} (x) = - \frac{1}{8 x} + \frac{1}{8}$