Algebra Examples

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

Step 1

Simplify $f^{- 1} x$ .

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2

Combine $\frac{1}{f}$ and $x$ .

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$f, f (x)$

Step 2.2

Since $f, f (x)$ contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $f, f (x)$ are:

1. Find the LCM for the numeric part $1, 1$ .

2. Find the LCM for the variable part $f^{1}, f^{1}$ .

3. Find the LCM for the compound variable part $x$ .

4. Multiply each LCM together.

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

The LCM of $1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.6

The factor for $f^{1}$ is $f$ itself.

$f^{1} = f$

$f$ occurs $1$ time.

Step 2.7

The LCM of $f^{1}, f^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$f$

Step 2.8

The factor for $x$ is $x$ itself.

$(x) = x$

$(x)$ occurs $1$ time.

Step 2.9

The LCM of $x$ is the result of multiplying all factors the greatest number of times they occur in either term.

$x$

Step 2.10

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$f (x)$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Combine $\frac{x}{f}$ and $f (x)$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Combine $\frac{1}{f (x)}$ and $f (x)$ .

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

Step 3.3.2

Cancel the common factor of $f (x)$ .

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 4

Solve the equation.

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

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

$f, 1$

Step 4.1.2

The LCM of one and any expression is the expression.

$f$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $f$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Rewrite the expression.

$x f (x) = 1 f$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Multiply $f$ by $1$ .

$x f (x) = f$

Step 4.3

Solve the equation.

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Step 4.3.1

Subtract $f$ from both sides of the equation.

$x f (x) - f = 0$

Step 4.3.2

Subtract $x f (x)$ from both sides of the equation.

$- f = - x f (x)$

Step 4.3.3

Divide each term in $- f = - x f (x)$ by $- 1$ and simplify.

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Step 4.3.3.1

Divide each term in $- f = - x f (x)$ by $- 1$ .

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

Step 4.3.3.2

Simplify the left side.

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Step 4.3.3.2.1

Dividing two negative values results in a positive value.

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

Step 4.3.3.2.2

Divide $f$ by $1$ .

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

Step 4.3.3.3

Simplify the right side.

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Step 4.3.3.3.1

Dividing two negative values results in a positive value.

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

Step 4.3.3.3.2

Divide $x f (x)$ by $1$ .

$f = x f (x)$

Step 4.3.4

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.4.1

Move $x$ .

$f = x \cdot x f$

Step 4.3.4.2

Multiply $x$ by $x$ .

$f = x^{2} f$

Step 4.3.5

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

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

Step 4.3.6

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

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Step 4.3.6.1

Factor $f$ out of $f^{1}$ .

$f \cdot 1 - x^{2} f = 0$

Step 4.3.6.2

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

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

Step 4.3.6.3

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

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

Step 4.3.7

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

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

Step 4.3.8

Factor.

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Step 4.3.8.1

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

$f ((1 + x) (1 - x)) = 0$

Step 4.3.8.2

Remove unnecessary parentheses.

$f (1 + x) (1 - x) = 0$

Step 4.3.9

Divide each term in $f (1 + x) (1 - x) = 0$ by $(1 + x) (1 - x)$ and simplify.

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Step 4.3.9.1

Divide each term in $f (1 + x) (1 - x) = 0$ by $(1 + x) (1 - x)$ .

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

Step 4.3.9.2

Simplify the left side.

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Step 4.3.9.2.1

Cancel the common factor of $1 + x$ .

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Step 4.3.9.2.1.1

Cancel the common factor.

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

Step 4.3.9.2.1.2

Rewrite the expression.

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

Step 4.3.9.2.2

Cancel the common factor of $1 - x$ .

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Step 4.3.9.2.2.1

Cancel the common factor.

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

Step 4.3.9.2.2.2

Divide $f$ by $1$ .

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

Step 4.3.9.3

Simplify the right side.

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Step 4.3.9.3.1

Divide $0$ by $(1 + x) (1 - x)$ .

$f = 0$