Finite Math Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Factor each term.

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Step 3.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

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

Step 3.2.3

Reduce the expression by cancelling the common factors.

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Step 3.2.3.1

Reduce the expression $\frac{(y + 1) (y - 1)}{y - 1}$ by cancelling the common factors.

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Step 3.2.3.1.1

Cancel the common factor.

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

Step 3.2.3.1.2

Rewrite the expression.

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

Step 3.2.3.2

Divide $y + 1$ by $1$ .

$y + 1 = x$

Step 3.3

Subtract $1$ from both sides of the equation.

$y = x - 1$

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.2.3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 5.2.3.2

Cancel the common factor of $x - 1$ .

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Step 5.2.3.2.1

Cancel the common factor.

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

Step 5.2.3.2.2

Divide $x + 1$ by $1$ .

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

Step 5.2.4

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

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Step 5.2.4.1

Subtract $1$ from $1$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

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

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

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.3.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x - 1$ and $b = 1$ .

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

Step 5.3.3.3

Simplify.

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Step 5.3.3.3.1

Add $- 1$ and $1$ .

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

Step 5.3.3.3.2

Add $x$ and $0$ .

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

Step 5.3.3.3.3

Subtract $1$ from $- 1$ .

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

Step 5.3.4

Reduce the expression by cancelling the common factors.

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Step 5.3.4.1

Subtract $1$ from $- 1$ .

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

Step 5.3.4.2

Cancel the common factor of $x - 2$ .

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Step 5.3.4.2.1

Cancel the common factor.

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

Step 5.3.4.2.2

Divide $x$ by $1$ .

$f (x - 1) = x$

Step 5.4

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

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