Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Add $2$ to both sides of the equation.

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

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

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

$y, 1, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$y$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $y$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Rewrite the expression.

$1 = x y + 2 y$

Step 3.5

Solve the equation.

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Step 3.5.1

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

$x y + 2 y = 1$

Step 3.5.2

Factor $y$ out of $x y + 2 y$ .

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Step 3.5.2.1

Factor $y$ out of $x y$ .

$y x + 2 y = 1$

Step 3.5.2.2

Factor $y$ out of $2 y$ .

$y x + y \cdot 2 = 1$

Step 3.5.2.3

Factor $y$ out of $y x + y \cdot 2$ .

$y (x + 2) = 1$

Step 3.5.3

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

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Step 3.5.3.1

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

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

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $x + 2$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

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

Step 3.5.3.2.1.2

Divide $y$ by $1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

Add $- 2$ and $2$ .

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

Step 5.2.3.2

Add $\frac{1}{x}$ and $0$ .

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

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.5

Multiply $x$ by $1$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.3.2

Multiply $x + 2$ by $1$ .

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

Step 5.3.4

Combine the opposite terms in $x + 2 - 2$ .

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Step 5.3.4.1

Subtract $2$ from $2$ .

$f (\frac{1}{x + 2}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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