Finite Math Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y^{2} - 1$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify $(y^{2} - 1) (x)$ .

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Step 3.2.1.1

Apply the distributive property.

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

Step 3.2.1.2

Rewrite $- 1 x$ as $- x$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $(\frac{y^{2}}{y^{2} - 1}) (y^{2} - 1)$ .

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Step 3.3.1.1

Simplify the denominator.

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Step 3.3.1.1.1

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

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

Step 3.3.1.1.2

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

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

Step 3.3.1.2

Multiply $\frac{y^{2}}{(y + 1) (y - 1)}$ by $y^{2} - 1$ .

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

Step 3.3.1.3

Simplify the numerator.

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Step 3.3.1.3.1

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

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

Step 3.3.1.3.2

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

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

Step 3.3.1.4

Reduce the expression by cancelling the common factors.

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Step 3.3.1.4.1

Cancel the common factor of $y + 1$ .

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Step 3.3.1.4.1.1

Cancel the common factor.

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

Step 3.3.1.4.1.2

Rewrite the expression.

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

Step 3.3.1.4.2

Cancel the common factor of $y - 1$ .

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Step 3.3.1.4.2.1

Cancel the common factor.

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

Step 3.3.1.4.2.2

Divide $y^{2}$ by $1$ .

$y^{2} x - x = y^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $y^{2}$ from both sides of the equation.

$y^{2} x - x - y^{2} = 0$

Step 3.4.2

Add $x$ to both sides of the equation.

$y^{2} x - y^{2} = x$

Step 3.4.3

Factor $y^{2}$ out of $y^{2} x - y^{2}$ .

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Step 3.4.3.1

Factor $y^{2}$ out of $y^{2} x$ .

$y^{2} (x) - y^{2} = x$

Step 3.4.3.2

Factor $y^{2}$ out of $- y^{2}$ .

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

Step 3.4.3.3

Factor $y^{2}$ out of $y^{2} (x) + y^{2} \cdot - 1$ .

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

Step 3.4.4

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

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Step 3.4.4.1

Divide each term in $y^{2} (x - 1) = x$ by $x - 1$ .

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

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $x - 1$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

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

Step 3.4.4.2.1.2

Divide $y^{2}$ by $1$ .

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

Step 3.4.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{x}{x - 1}}$

Step 3.4.6

Simplify $\pm \sqrt{\frac{x}{x - 1}}$ .

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Step 3.4.6.1

Rewrite $\sqrt{\frac{x}{x - 1}}$ as $\frac{\sqrt{x}}{\sqrt{x - 1}}$ .

$y = \pm \frac{\sqrt{x}}{\sqrt{x - 1}}$

Step 3.4.6.2

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

$y = \pm \frac{\sqrt{x}}{\sqrt{x - 1}} \cdot \frac{\sqrt{x - 1}}{\sqrt{x - 1}}$

Step 3.4.6.3

Combine and simplify the denominator.

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Step 3.4.6.3.1

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

$y = \pm \frac{\sqrt{x} \sqrt{x - 1}}{\sqrt{x - 1} \sqrt{x - 1}}$

Step 3.4.6.3.2

Raise $\sqrt{x - 1}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x} \sqrt{x - 1}}{{\sqrt{x - 1}}^{1} \sqrt{x - 1}}$

Step 3.4.6.3.3

Raise $\sqrt{x - 1}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x} \sqrt{x - 1}}{{\sqrt{x - 1}}^{1} {\sqrt{x - 1}}^{1}}$

Step 3.4.6.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{x} \sqrt{x - 1}}{{\sqrt{x - 1}}^{1 + 1}}$

Step 3.4.6.3.5

Add $1$ and $1$ .

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

Step 3.4.6.3.6

Rewrite ${\sqrt{x - 1}}^{2}$ as $x - 1$ .

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Step 3.4.6.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 1}$ as ${(x - 1)}^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x} \sqrt{x - 1}}{{({(x - 1)}^{\frac{1}{2}})}^{2}}$

Step 3.4.6.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{1}{2} \cdot 2}}$

Step 3.4.6.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{2}{2}}}$

Step 3.4.6.3.6.4

Cancel the common factor of $2$ .

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Step 3.4.6.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{2}{2}}}$

Step 3.4.6.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{1}}$

Step 3.4.6.3.6.5

Simplify.

$y = \pm \frac{\sqrt{x} \sqrt{x - 1}}{x - 1}$

Step 3.4.6.4

Combine using the product rule for radicals.

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

Step 3.4.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.7.1

First, use the positive value of the $\pm$ to find the first solution.