Finite Math Examples

$\frac{7 x - 14}{x - 2}$

Step 1

Interchange the variables.

$x = \frac{7 y - 14}{y - 2}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{7 y - 14}{y - 2} = x$ .

$\frac{7 y - 14}{y - 2} = x$

Step 2.2

Multiply both sides by $y - 2$ .

$\frac{7 y - 14}{y - 2} (y - 2) = x (y - 2)$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $y - 2$ .

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Step 2.3.1.1.1

Cancel the common factor.

$\frac{7 y - 14}{y - 2} (y - 2) = x (y - 2)$

Step 2.3.1.1.2

Rewrite the expression.

$7 y - 14 = x (y - 2)$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $x (y - 2)$ .

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Step 2.3.2.1.1

Apply the distributive property.

$7 y - 14 = x y + x \cdot - 2$

Step 2.3.2.1.2

Move $- 2$ to the left of $x$ .

$7 y - 14 = x y - 2 x$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Subtract $x y$ from both sides of the equation.

$7 y - 14 - x y = - 2 x$

Step 2.4.2

Add $14$ to both sides of the equation.

$7 y - x y = - 2 x + 14$

Step 2.4.3

Factor $y$ out of $7 y - x y$ .

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Step 2.4.3.1

Factor $y$ out of $7 y$ .

$y \cdot 7 - x y = - 2 x + 14$

Step 2.4.3.2

Factor $y$ out of $- x y$ .

$y \cdot 7 + y (- x) = - 2 x + 14$

Step 2.4.3.3

Factor $y$ out of $y \cdot 7 + y (- x)$ .

$y (7 - x) = - 2 x + 14$

Step 2.4.4

Divide each term in $y (7 - x) = - 2 x + 14$ by $7 - x$ and simplify.

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Step 2.4.4.1

Divide each term in $y (7 - x) = - 2 x + 14$ by $7 - x$ .

$\frac{y (7 - x)}{7 - x} = \frac{- 2 x}{7 - x} + \frac{14}{7 - x}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $7 - x$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{y (7 - x)}{7 - x} = \frac{- 2 x}{7 - x} + \frac{14}{7 - x}$

Step 2.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 2 x}{7 - x} + \frac{14}{7 - x}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Combine the numerators over the common denominator.

$y = \frac{- 2 x + 14}{7 - x}$

Step 2.4.4.3.2

Factor $2$ out of $- 2 x + 14$ .

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Step 2.4.4.3.2.1

Factor $2$ out of $- 2 x$ .

$y = \frac{2 (- x) + 14}{7 - x}$

Step 2.4.4.3.2.2

Factor $2$ out of $14$ .

$y = \frac{2 (- x) + 2 (7)}{7 - x}$

Step 2.4.4.3.2.3

Factor $2$ out of $2 (- x) + 2 (7)$ .

$y = \frac{2 (- x + 7)}{7 - x}$

Step 2.4.4.3.3

Cancel the common factor of $- x + 7$ and $7 - x$ .

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Step 2.4.4.3.3.1

Reorder terms.

$y = \frac{2 (- x + 7)}{- x + 7}$

Step 2.4.4.3.3.2

Cancel the common factor.

$y = \frac{2 (- x + 7)}{- x + 7}$

Step 2.4.4.3.3.3

Divide $2$ by $1$ .

$y = 2$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = 2$ is the inverse of $f (x) = \frac{7 x - 14}{x - 2}$ .

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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 $f^{- 1} (\frac{7 x - 14}{x - 2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7 x - 14}{x - 2}) = 2$

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

$f (2) = \frac{7 (2) - 14}{(2) - 2}$

Step 4.3.3

Subtract $2$ from $2$ .

$f (2) = \frac{7 (2) - 14}{0}$

Step 4.3.4

The expression contains a division by $0$ . The expression is undefined.

$f (2) = Undefined$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 2$ is the inverse of $f (x) = \frac{7 x - 14}{x - 2}$ .

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