Precalculus Examples

$f (x) = \frac{1}{x + 7}$

Step 1

Write $f (x) = \frac{1}{x + 7}$ as an equation.

$y = \frac{1}{x + 7}$

Step 2

Interchange the variables.

$x = \frac{1}{y + 7}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{y + 7} = x$ .

$\frac{1}{y + 7} = x$

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.

$y + 7, 1$

Step 3.2.2

Remove parentheses.

$y + 7, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$y + 7$

Step 3.3

Multiply each term in $\frac{1}{y + 7} = x$ by $y + 7$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{1}{y + 7} = x$ by $y + 7$ .

$\frac{1}{y + 7} (y + 7) = x (y + 7)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y + 7$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{1}{y + 7} (y + 7) = x (y + 7)$

Step 3.3.2.1.2

Rewrite the expression.

$1 = x (y + 7)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

$1 = x y + x \cdot 7$

Step 3.3.3.2

Move $7$ to the left of $x$ .

$1 = x y + 7 x$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $x y + 7 x = 1$ .

$x y + 7 x = 1$

Step 3.4.2

Subtract $7 x$ from both sides of the equation.

$x y = 1 - 7 x$

Step 3.4.3

Divide each term in $x y = 1 - 7 x$ by $x$ and simplify.

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Step 3.4.3.1

Divide each term in $x y = 1 - 7 x$ by $x$ .

$\frac{x y}{x} = \frac{1}{x} + \frac{- 7 x}{x}$

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Cancel the common factor of $x$ .

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Step 3.4.3.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{1}{x} + \frac{- 7 x}{x}$

Step 3.4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{x} + \frac{- 7 x}{x}$

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

Cancel the common factor of $x$ .

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Step 3.4.3.3.1.1

Cancel the common factor.

$y = \frac{1}{x} + \frac{- 7 x}{x}$

Step 3.4.3.3.1.2

Divide $- 7$ by $1$ .

$y = \frac{1}{x} - 7$

Step 4

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

$f^{- 1} (x) = \frac{1}{x} - 7$

Step 5

Verify if $f^{- 1} (x) = \frac{1}{x} - 7$ is the inverse of $f (x) = \frac{1}{x + 7}$ .

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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 $f^{- 1} (\frac{1}{x + 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{x + 7}) = \frac{1}{\frac{1}{x + 7}} - 7$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{1}{x + 7}) = 1 (x + 7) - 7$

Step 5.2.3.2

Multiply $x + 7$ by $1$ .

$f^{- 1} (\frac{1}{x + 7}) = x + 7 - 7$

Step 5.2.4

Combine the opposite terms in $x + 7 - 7$ .

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Step 5.2.4.1

Subtract $7$ from $7$ .

$f^{- 1} (\frac{1}{x + 7}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{1}{x + 7}) = x$

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 $f (\frac{1}{x} - 7)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{1}{x} - 7) = \frac{1}{(\frac{1}{x} - 7) + 7}$

Step 5.3.3

Simplify the denominator.

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Step 5.3.3.1

Add $- 7$ and $7$ .

$f (\frac{1}{x} - 7) = \frac{1}{\frac{1}{x} + 0}$

Step 5.3.3.2

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

$f (\frac{1}{x} - 7) = \frac{1}{\frac{1}{x}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{1}{x} - 7) = 1 x$

Step 5.3.5

Multiply $x$ by $1$ .

$f (\frac{1}{x} - 7) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{1}{x} - 7$ is the inverse of $f (x) = \frac{1}{x + 7}$ .

$f^{- 1} (x) = \frac{1}{x} - 7$