Finite Math Examples

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

Step 1

Simplify $f (f^{- 1} x)$ .

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Step 1.1

Multiply $f$ by $f^{- 1}$ by adding the exponents.

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Step 1.1.1

Move $f^{- 1}$ .

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

Step 1.1.2

Multiply $f^{- 1}$ by $f$ .

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Step 1.1.2.1

Raise $f$ to the power of $1$ .

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

Step 1.1.2.2

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

$f^{- 1 + 1} x = f^{- 1}$

Step 1.1.3

Add $- 1$ and $1$ .

$f^{0} x = f^{- 1}$

Step 1.2

Simplify $f^{0} x$ .

$x = f^{- 1}$

Step 2

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

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

Step 3

Since $f$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 4

Find the LCD of the terms in the equation.

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Step 4.1

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

$f, 1$

Step 4.2

The LCM of one and any expression is the expression.

$f$

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $f$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Rewrite the expression.

$1 = x f$

Step 6

Solve the equation.

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Step 6.1

Rewrite the equation as $x f = 1$ .

$x f = 1$

Step 6.2

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

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Step 6.2.1

Divide each term in $x f = 1$ by $x$ .

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $x$ .

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Step 6.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.1.2

Divide $f$ by $1$ .

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