Algebra Examples

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

Step 1

Divide each term in $f^{- 1} x = x$ by $x$ and simplify.

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Step 1.1

Divide each term in $f^{- 1} x = x$ by $x$ .

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Move $f^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2.2

Cancel the common factor of $x$ .

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $x$ .

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Step 1.3.1.1

Cancel the common factor.

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

Step 1.3.1.2

Rewrite the expression.

$\frac{1}{f} = 1$

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.

$f, 1$

Step 2.2

The LCM of one and any expression is the expression.

$f$

Step 3

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

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Step 3.1

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

$\frac{1}{f} f = 1 f$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $f$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{1}{f} f = 1 f$

Step 3.2.1.2

Rewrite the expression.

$1 = 1 f$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $f$ by $1$ .

$1 = f$

Step 4

Rewrite the equation as $f = 1$ .

$f = 1$

Step 5

To rewrite as a function of $x$ , write the equation so that $f$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = 1$