Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Factor each term.

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Step 3.2.1

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

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

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 $b = 1$ .

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

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

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

$(y + 1) (y - 1), 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$(y + 1) (y - 1)$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $(y + 1) (y - 1)$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Rewrite the expression.

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Expand $(y + 1) (y - 1)$ using the FOIL Method.

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Step 3.4.3.1.1

Apply the distributive property.

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

Step 3.4.3.1.2

Apply the distributive property.

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

Step 3.4.3.1.3

Apply the distributive property.

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

Step 3.4.3.2

Simplify and combine like terms.

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Step 3.4.3.2.1

Simplify each term.

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Step 3.4.3.2.1.1

Multiply $y$ by $y$ .

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

Step 3.4.3.2.1.2

Move $- 1$ to the left of $y$ .

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

Step 3.4.3.2.1.3

Rewrite $- 1 y$ as $- y$ .

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

Step 3.4.3.2.1.4

Multiply $y$ by $1$ .

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

Step 3.4.3.2.1.5

Multiply $- 1$ by $1$ .

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

Step 3.4.3.2.2

Add $- y$ and $y$ .

$1 = x (y^{2} + 0 - 1)$

Step 3.4.3.2.3

Add $y^{2}$ and $0$ .

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

Step 3.4.3.3

Simplify by multiplying through.

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Step 3.4.3.3.1

Apply the distributive property.

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

Step 3.4.3.3.2

Move $- 1$ to the left of $x$ .

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

Step 3.4.3.4

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

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

Step 3.5

Solve the equation.

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Step 3.5.1

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

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

Step 3.5.2

Add $x$ to both sides of the equation.

$x y^{2} = 1 + x$

Step 3.5.3

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

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Step 3.5.3.1

Divide each term in $x y^{2} = 1 + x$ by $x$ .

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

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $x$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

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

Step 3.5.3.2.1.2

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

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

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Cancel the common factor of $x$ .

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Step 3.5.3.3.1.1

Cancel the common factor.

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

Step 3.5.3.3.1.2

Rewrite the expression.

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

Step 3.5.4

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

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

Step 3.5.5

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

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Step 3.5.5.1

Write $1$ as a fraction with a common denominator.

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

Step 3.5.5.2

Combine the numerators over the common denominator.

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

Step 3.5.5.3

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

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

Step 3.5.5.4

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

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

Step 3.5.5.5

Combine and simplify the denominator.

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Step 3.5.5.5.1

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

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

Step 3.5.5.5.2

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

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

Step 3.5.5.5.3

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

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

Step 3.5.5.5.4

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

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

Step 3.5.5.5.5

Add $1$ and $1$ .

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

Step 3.5.5.5.6

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

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Step 3.5.5.5.6.1

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

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

Step 3.5.5.5.6.2

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

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

Step 3.5.5.5.6.3

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

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

Step 3.5.5.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.5.5.6.4.1

Cancel the common factor.

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

Step 3.5.5.5.6.4.2

Rewrite the expression.

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

Step 3.5.5.5.6.5

Simplify.

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

Step 3.5.5.6

Combine using the product rule for radicals.

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

Step 3.5.5.7

Reorder factors in $\pm \frac{\sqrt{(1 + x) x}}{x}$ .

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

Step 3.5.6

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

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Step 3.5.6.1

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

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

Step 3.5.6.2

Next, use the negative value of the $\pm$ to find the second solution.