Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

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 $\frac{y - 7}{y + 7} = x$ by $y + 7$ to eliminate the fractions.

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Step 3.3.1

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

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

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.

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

Step 3.3.2.1.2

Rewrite the expression.

$y - 7 = x (y + 7)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

$y - 7 = x y + x \cdot 7$

Step 3.3.3.2

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

$y - 7 = x y + 7 x$

Step 3.4

Solve the equation.

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Step 3.4.1

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

$y - 7 - x y = 7 x$

Step 3.4.2

Add $7$ to both sides of the equation.

$y - x y = 7 x + 7$

Step 3.4.3

Factor $y$ out of $y - x y$ .

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Step 3.4.3.1

Factor $y$ out of $y^{1}$ .

$y \cdot 1 - x y = 7 x + 7$

Step 3.4.3.2

Factor $y$ out of $- x y$ .

$y \cdot 1 + y (- x) = 7 x + 7$

Step 3.4.3.3

Factor $y$ out of $y \cdot 1 + y (- x)$ .

$y (1 - x) = 7 x + 7$

Step 3.4.4

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

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Step 3.4.4.1

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

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

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $1 - x$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

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

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Combine the numerators over the common denominator.

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

Step 3.4.4.3.2

Factor $7$ out of $7 x + 7$ .

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Step 3.4.4.3.2.1

Factor $7$ out of $7 x$ .

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

Step 3.4.4.3.2.2

Factor $7$ out of $7$ .

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

Step 3.4.4.3.2.3

Factor $7$ out of $7 (x) + 7 (1)$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Write $1$ as a fraction with a common denominator.

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

Step 5.2.3.2

Combine the numerators over the common denominator.

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

Step 5.2.3.3

Reorder terms.

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

Step 5.2.3.4

Rewrite $\frac{x + x - 7 + 7}{x + 7}$ in a factored form.

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Step 5.2.3.4.1

Add $x$ and $x$ .

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{7 (\frac{2 x - 7 + 7}{x + 7})}{1 - \frac{x - 7}{x + 7}}$

Step 5.2.3.4.2

Add $- 7$ and $7$ .

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{7 (\frac{2 x + 0}{x + 7})}{1 - \frac{x - 7}{x + 7}}$

Step 5.2.3.4.3

Add $2 x$ and $0$ .

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{7 (\frac{2 x}{x + 7})}{1 - \frac{x - 7}{x + 7}}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Write $1$ as a fraction with a common denominator.

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{7 (\frac{2 x}{x + 7})}{\frac{x + 7}{x + 7} - \frac{x - 7}{x + 7}}$

Step 5.2.4.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{7 (\frac{2 x}{x + 7})}{\frac{x + 7 - (x - 7)}{x + 7}}$

Step 5.2.4.3

Rewrite $\frac{x + 7 - (x - 7)}{x + 7}$ in a factored form.

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Step 5.2.4.3.1

Apply the distributive property.

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{7 (\frac{2 x}{x + 7})}{\frac{x + 7 - x + 7}{x + 7}}$

Step 5.2.4.3.2

Multiply $- 1$ by $- 7$ .

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{7 (\frac{2 x}{x + 7})}{\frac{x + 7 - x + 7}{x + 7}}$

Step 5.2.4.3.3

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{7 (\frac{2 x}{x + 7})}{\frac{0 + 7 + 7}{x + 7}}$

Step 5.2.4.3.4

Add $0$ and $7$ .

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{7 (\frac{2 x}{x + 7})}{\frac{7 + 7}{x + 7}}$

Step 5.2.4.3.5

Add $7$ and $7$ .

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{7 (\frac{2 x}{x + 7})}{\frac{14}{x + 7}}$

Step 5.2.5

Combine $7$ and $\frac{2 x}{x + 7}$ .

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{\frac{7 (2 x)}{x + 7}}{\frac{14}{x + 7}}$

Step 5.2.6

Multiply $7$ by $2$ .

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{\frac{14 x}{x + 7}}{\frac{14}{x + 7}}$

Step 5.2.7

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{14 x}{x + 7} \cdot \frac{x + 7}{14}$

Step 5.2.8

Cancel the common factor of $14$ .

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Step 5.2.8.1

Factor $14$ out of $14 x$ .

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{14 (x)}{x + 7} \cdot \frac{x + 7}{14}$

Step 5.2.8.2

Cancel the common factor.

$f^{- 1} (\frac{x - 7}{x + 7}) = \frac{14 x}{x + 7} \cdot \frac{x + 7}{14}$

Step 5.2.8.3

Rewrite the expression.

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

Step 5.2.9

Cancel the common factor of $x + 7$ .

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Step 5.2.9.1

Cancel the common factor.

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

Step 5.2.9.2

Rewrite the expression.

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

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

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

Step 5.3.3

Cancel the common factor of $(\frac{7 (x + 1)}{1 - x}) - 7$ and $(\frac{7 (x + 1)}{1 - x}) + 7$ .

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Step 5.3.3.1

Factor $7$ out of $\frac{7 (x + 1)}{1 - x}$ .

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

Step 5.3.3.2

Factor $7$ out of $- 7$ .

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

Step 5.3.3.3

Factor $7$ out of $7 \frac{x + 1}{1 - x} + 7 (- 1)$ .

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

Step 5.3.3.4

Cancel the common factors.

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Step 5.3.3.4.1

Factor $7$ out of $\frac{7 (x + 1)}{1 - x}$ .

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

Step 5.3.3.4.2

Factor $7$ out of $7$ .

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

Step 5.3.3.4.3

Factor $7$ out of $7 \frac{x + 1}{1 - x} + 7 (1)$ .

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

Step 5.3.3.4.4

Cancel the common factor.

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

Step 5.3.3.4.5

Rewrite the expression.

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

Step 5.3.4

Multiply the numerator and denominator of the fraction by $1 - x$ .

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Step 5.3.4.1

Multiply $\frac{\frac{x + 1}{1 - x} - 1}{\frac{x + 1}{1 - x} + 1}$ by $\frac{1 - x}{1 - x}$ .

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

Step 5.3.4.2

Combine.

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

Step 5.3.5

Apply the distributive property.

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

Step 5.3.6

Simplify by cancelling.

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Step 5.3.6.1

Cancel the common factor of $1 - x$ .

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Step 5.3.6.1.1

Cancel the common factor.

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

Step 5.3.6.1.2

Rewrite the expression.

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

Step 5.3.6.2

Cancel the common factor of $1 - x$ .

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Step 5.3.6.2.1

Cancel the common factor.

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

Step 5.3.6.2.2

Rewrite the expression.

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

Step 5.3.7

Simplify the numerator.

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Step 5.3.7.1

Apply the distributive property.

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

Step 5.3.7.2

Multiply $- 1$ by $1$ .

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

Step 5.3.7.3

Multiply $- x \cdot - 1$ .

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Step 5.3.7.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.7.3.2

Multiply $x$ by $1$ .

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

Step 5.3.7.4

Add $x$ and $x$ .

$f (\frac{7 (x + 1)}{1 - x}) = \frac{2 x + 1 - 1}{x + 1 + (1 - x) \cdot 1}$

Step 5.3.7.5

Subtract $1$ from $1$ .

$f (\frac{7 (x + 1)}{1 - x}) = \frac{2 x + 0}{x + 1 + (1 - x) \cdot 1}$

Step 5.3.7.6

Add $2 x$ and $0$ .

$f (\frac{7 (x + 1)}{1 - x}) = \frac{2 x}{x + 1 + (1 - x) \cdot 1}$

Step 5.3.8

Simplify the denominator.

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Step 5.3.8.1

Multiply $1 - x$ by $1$ .

$f (\frac{7 (x + 1)}{1 - x}) = \frac{2 x}{x + 1 + 1 - x}$

Step 5.3.8.2

Subtract $x$ from $x$ .

$f (\frac{7 (x + 1)}{1 - x}) = \frac{2 x}{0 + 1 + 1}$

Step 5.3.8.3

Add $0$ and $1$ .

$f (\frac{7 (x + 1)}{1 - x}) = \frac{2 x}{1 + 1}$

Step 5.3.8.4

Add $1$ and $1$ .

$f (\frac{7 (x + 1)}{1 - x}) = \frac{2 x}{2}$

Step 5.3.9

Cancel the common factor of $2$ .

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Step 5.3.9.1

Cancel the common factor.

$f (\frac{7 (x + 1)}{1 - x}) = \frac{2 x}{2}$

Step 5.3.9.2

Divide $x$ by $1$ .

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

Step 5.4

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

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