Precalculus Examples

$\frac{1}{x}$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$y, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$y$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $y$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Rewrite the expression.

$1 = x y$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $x y = 1$ .

$x y = 1$

Step 2.4.2

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

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Step 2.4.2.1

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

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $x$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.1.2

Divide $y$ by $1$ .

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

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

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

Step 4

Verify if $f^{- 1} (x) = \frac{1}{x}$ is the inverse of $f (x) = \frac{1}{x}$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\frac{1}{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 4.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.4

Multiply $x$ by $1$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{1}{x})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.4

Multiply $x$ by $1$ .

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

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{1}{x}$ is the inverse of $f (x) = \frac{1}{x}$ .

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