Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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

$2 y, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$2 y$

Step 3.3

Multiply each term in $\frac{1}{2 y} = x$ by $2 y$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{1}{2 y} = x$ by $2 y$ .

$\frac{1}{2 y} (2 y) = x (2 y)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Rewrite using the commutative property of multiplication.

$2 \frac{1}{2 y} y = x (2 y)$

Step 3.3.2.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1

Factor $2$ out of $2 y$ .

$2 \frac{1}{2 (y)} y = x (2 y)$

Step 3.3.2.2.2

Cancel the common factor.

$2 \frac{1}{2 y} y = x (2 y)$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{1}{y} y = x (2 y)$

Step 3.3.2.3

Cancel the common factor of $y$ .

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Step 3.3.2.3.1

Cancel the common factor.

$\frac{1}{y} y = x (2 y)$

Step 3.3.2.3.2

Rewrite the expression.

$1 = x (2 y)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Rewrite using the commutative property of multiplication.

$1 = 2 x y$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $2 x y = 1$ .

$2 x y = 1$

Step 3.4.2

Divide each term in $2 x y = 1$ by $2 x$ and simplify.

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Step 3.4.2.1

Divide each term in $2 x y = 1$ by $2 x$ .

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

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

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

Step 3.4.2.2.1.2

Rewrite the expression.

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

Step 3.4.2.2.2

Cancel the common factor of $x$ .

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Step 3.4.2.2.2.1

Cancel the common factor.

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

Step 3.4.2.2.2.2

Divide $y$ by $1$ .

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{1}{2 x}$ is the inverse of $f (x) = \frac{1}{2 x}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{1}{2 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

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

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

Step 5.2.4

Reduce the expression $\frac{2}{2 x}$ by cancelling the common factors.

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Step 5.2.4.1

Cancel the common factor.

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

Step 5.2.4.2

Rewrite the expression.

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

Step 5.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.6

Multiply $x$ by $1$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{1}{2 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 5.3.3

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

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

Step 5.3.4

Reduce the expression $\frac{2}{2 x}$ by cancelling the common factors.

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Step 5.3.4.1

Cancel the common factor.

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

Step 5.3.4.2

Rewrite the expression.

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

Step 5.3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.6

Multiply $x$ by $1$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{1}{2 x}$ is the inverse of $f (x) = \frac{1}{2 x}$ .

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