Precalculus Examples

$f (x) = \frac{x - 6}{x + 5}$

Step 1

Write $f (x) = \frac{x - 6}{x + 5}$ as an equation.

$y = \frac{x - 6}{x + 5}$

Step 2

Interchange the variables.

$x = \frac{y - 6}{y + 5}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y - 6}{y + 5} = x$ .

$\frac{y - 6}{y + 5} = x$

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.

$y + 5, 1$

Step 3.2.2

Remove parentheses.

$y + 5, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$y + 5$

Step 3.3

Multiply each term in $\frac{y - 6}{y + 5} = x$ by $y + 5$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{y - 6}{y + 5} = x$ by $y + 5$ .

$\frac{y - 6}{y + 5} (y + 5) = x (y + 5)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y + 5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{y - 6}{y + 5} (y + 5) = x (y + 5)$

Step 3.3.2.1.2

Rewrite the expression.

$y - 6 = x (y + 5)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

$y - 6 = x y + x \cdot 5$

Step 3.3.3.2

Move $5$ to the left of $x$ .

$y - 6 = x y + 5 x$

Step 3.4

Solve the equation.

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Step 3.4.1

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

$y - 6 - x y = 5 x$

Step 3.4.2

Add $6$ to both sides of the equation.

$y - x y = 5 x + 6$

Step 3.4.3

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

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Step 3.4.3.1

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

$y \cdot 1 - x y = 5 x + 6$

Step 3.4.3.2

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

$y \cdot 1 + y (- x) = 5 x + 6$

Step 3.4.3.3

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

$y (1 - x) = 5 x + 6$

Step 3.4.4

Divide each term in $y (1 - x) = 5 x + 6$ by $1 - x$ and simplify.

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Step 3.4.4.1

Divide each term in $y (1 - x) = 5 x + 6$ by $1 - x$ .

$\frac{y (1 - x)}{1 - x} = \frac{5 x}{1 - x} + \frac{6}{1 - x}$

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.

$\frac{y (1 - x)}{1 - x} = \frac{5 x}{1 - x} + \frac{6}{1 - x}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 x}{1 - x} + \frac{6}{1 - x}$

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.

$y = \frac{5 x + 6}{1 - x}$

Step 4

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

$f^{- 1} (x) = \frac{5 x + 6}{1 - x}$

Step 5

Verify if $f^{- 1} (x) = \frac{5 x + 6}{1 - x}$ is the inverse of $f (x) = \frac{x - 6}{x + 5}$ .

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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 $f^{- 1} (\frac{x - 6}{x + 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{5 (\frac{x - 6}{x + 5}) + 6}{1 - (\frac{x - 6}{x + 5})}$

Step 5.2.3

Multiply the numerator and denominator of the fraction by $x + 5$ .

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Step 5.2.3.1

Multiply $\frac{5 \frac{x - 6}{x + 5} + 6}{1 - \frac{x - 6}{x + 5}}$ by $\frac{x + 5}{x + 5}$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{x + 5}{x + 5} \cdot \frac{5 (\frac{x - 6}{x + 5}) + 6}{1 - \frac{x - 6}{x + 5}}$

Step 5.2.3.2

Combine.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (5 (\frac{x - 6}{x + 5}) + 6)}{(x + 5) (1 - \frac{x - 6}{x + 5})}$

Step 5.2.4

Apply the distributive property.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (5 (\frac{x - 6}{x + 5})) + (x + 5) \cdot 6}{(x + 5) \cdot 1 + (x + 5) (- \frac{x - 6}{x + 5})}$

Step 5.2.5

Cancel the common factor of $x + 5$ .

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Step 5.2.5.1

Move the leading negative in $- \frac{x - 6}{x + 5}$ into the numerator.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (5 (\frac{x - 6}{x + 5})) + (x + 5) \cdot 6}{(x + 5) \cdot 1 + (x + 5) (\frac{- (x - 6)}{x + 5})}$

Step 5.2.5.2

Cancel the common factor.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (5 (\frac{x - 6}{x + 5})) + (x + 5) \cdot 6}{(x + 5) \cdot 1 + (x + 5) (\frac{- (x - 6)}{x + 5})}$

Step 5.2.5.3

Rewrite the expression.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (5 (\frac{x - 6}{x + 5})) + (x + 5) \cdot 6}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Factor $x + 5$ out of $(x + 5) \cdot 5 \frac{x - 6}{x + 5} + (x + 5) \cdot 6$ .

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Step 5.2.6.1.1

Factor $x + 5$ out of $(x + 5) \cdot 5 \frac{x - 6}{x + 5}$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (5 (\frac{x - 6}{x + 5})) + (x + 5) \cdot 6}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.1.2

Factor $x + 5$ out of $(x + 5) \cdot 6$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (5 (\frac{x - 6}{x + 5})) + (x + 5) (6)}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.1.3

Factor $x + 5$ out of $(x + 5) (5 \frac{x - 6}{x + 5}) + (x + 5) (6)$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (5 (\frac{x - 6}{x + 5}) + 6)}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.2

Combine $5$ and $\frac{x - 6}{x + 5}$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{5 (x - 6)}{x + 5} + 6)}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.3

To write $6$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{5 (x - 6)}{x + 5} + \frac{6 (x + 5)}{x + 5})}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{5 (x - 6) + 6 (x + 5)}{x + 5})}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.5

Reorder terms.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{6 (x + 5) + 5 (x - 6)}{x + 5})}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.6

Rewrite $\frac{6 (x + 5) + 5 (x - 6)}{x + 5}$ in a factored form.

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Step 5.2.6.6.1

Apply the distributive property.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{6 x + 6 \cdot 5 + 5 (x - 6)}{x + 5})}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.6.2

Multiply $6$ by $5$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{6 x + 30 + 5 (x - 6)}{x + 5})}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.6.3

Apply the distributive property.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{6 x + 30 + 5 x + 5 \cdot - 6}{x + 5})}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.6.4

Multiply $5$ by $- 6$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{6 x + 30 + 5 x - 30}{x + 5})}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.6.5

Add $6 x$ and $5 x$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{11 x + 30 - 30}{x + 5})}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.6.6

Subtract $30$ from $30$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{11 x + 0}{x + 5})}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.6.6.7

Add $11 x$ and $0$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{11 x}{x + 5})}{(x + 5) \cdot 1 - (x - 6)}$

Step 5.2.7

Simplify the denominator.

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Step 5.2.7.1

Multiply $x + 5$ by $1$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{11 x}{x + 5})}{x + 5 - (x - 6)}$

Step 5.2.7.2

Apply the distributive property.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{11 x}{x + 5})}{x + 5 - x + 6}$

Step 5.2.7.3

Multiply $- 1$ by $- 6$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{11 x}{x + 5})}{x + 5 - x + 6}$

Step 5.2.7.4

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{11 x}{x + 5})}{0 + 5 + 6}$

Step 5.2.7.5

Add $0$ and $5$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{11 x}{x + 5})}{5 + 6}$

Step 5.2.7.6

Add $5$ and $6$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{(x + 5) (\frac{11 x}{x + 5})}{11}$

Step 5.2.8

Reduce the expression by cancelling the common factors.

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Step 5.2.8.1

Factor $\frac{11 x}{x + 5}$ out of $\frac{(x + 5) \frac{11 x}{x + 5}}{11}$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{11 x}{x + 5} \cdot \frac{x + 5}{11}$

Step 5.2.8.2

Cancel the common factor of $11$ .

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Step 5.2.8.2.1

Factor $11$ out of $11 x$ .

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{11 (x)}{x + 5} \cdot \frac{x + 5}{11}$

Step 5.2.8.2.2

Cancel the common factor.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{11 x}{x + 5} \cdot \frac{x + 5}{11}$

Step 5.2.8.2.3

Rewrite the expression.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{x}{x + 5} \cdot (x + 5)$

Step 5.2.8.3

Cancel the common factor of $x + 5$ .

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Step 5.2.8.3.1

Cancel the common factor.

$f^{- 1} (\frac{x - 6}{x + 5}) = \frac{x}{x + 5} \cdot (x + 5)$

Step 5.2.8.3.2

Rewrite the expression.

$f^{- 1} (\frac{x - 6}{x + 5}) = x$

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 $f (\frac{5 x + 6}{1 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{(\frac{5 x + 6}{1 - x}) - 6}{(\frac{5 x + 6}{1 - x}) + 5}$

Step 5.3.3

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

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Step 5.3.3.1

Multiply $\frac{\frac{5 x + 6}{1 - x} - 6}{\frac{5 x + 6}{1 - x} + 5}$ by $\frac{1 - x}{1 - x}$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{1 - x}{1 - x} \cdot \frac{\frac{5 x + 6}{1 - x} - 6}{\frac{5 x + 6}{1 - x} + 5}$

Step 5.3.3.2

Combine.

$f (\frac{5 x + 6}{1 - x}) = \frac{(1 - x) (\frac{5 x + 6}{1 - x} - 6)}{(1 - x) (\frac{5 x + 6}{1 - x} + 5)}$

Step 5.3.4

Apply the distributive property.

$f (\frac{5 x + 6}{1 - x}) = \frac{(1 - x) (\frac{5 x + 6}{1 - x}) + (1 - x) \cdot - 6}{(1 - x) (\frac{5 x + 6}{1 - x}) + (1 - x) \cdot 5}$

Step 5.3.5

Simplify by cancelling.

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Step 5.3.5.1

Cancel the common factor of $1 - x$ .

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Step 5.3.5.1.1

Cancel the common factor.

$f (\frac{5 x + 6}{1 - x}) = \frac{(1 - x) (\frac{5 x + 6}{1 - x}) + (1 - x) \cdot - 6}{(1 - x) (\frac{5 x + 6}{1 - x}) + (1 - x) \cdot 5}$

Step 5.3.5.1.2

Rewrite the expression.

$f (\frac{5 x + 6}{1 - x}) = \frac{5 x + 6 + (1 - x) \cdot - 6}{(1 - x) (\frac{5 x + 6}{1 - x}) + (1 - x) \cdot 5}$

Step 5.3.5.2

Cancel the common factor of $1 - x$ .

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Step 5.3.5.2.1

Cancel the common factor.

$f (\frac{5 x + 6}{1 - x}) = \frac{5 x + 6 + (1 - x) \cdot - 6}{(1 - x) (\frac{5 x + 6}{1 - x}) + (1 - x) \cdot 5}$

Step 5.3.5.2.2

Rewrite the expression.

$f (\frac{5 x + 6}{1 - x}) = \frac{5 x + 6 + (1 - x) \cdot - 6}{5 x + 6 + (1 - x) \cdot 5}$

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Apply the distributive property.

$f (\frac{5 x + 6}{1 - x}) = \frac{5 x + 6 + 1 \cdot - 6 - x \cdot - 6}{5 x + 6 + (1 - x) \cdot 5}$

Step 5.3.6.2

Multiply $- 6$ by $1$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{5 x + 6 - 6 - x \cdot - 6}{5 x + 6 + (1 - x) \cdot 5}$

Step 5.3.6.3

Multiply $- 6$ by $- 1$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{5 x + 6 - 6 + 6 x}{5 x + 6 + (1 - x) \cdot 5}$

Step 5.3.6.4

Add $5 x$ and $6 x$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{11 x + 6 - 6}{5 x + 6 + (1 - x) \cdot 5}$

Step 5.3.6.5

Subtract $6$ from $6$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{11 x + 0}{5 x + 6 + (1 - x) \cdot 5}$

Step 5.3.6.6

Add $11 x$ and $0$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{11 x}{5 x + 6 + (1 - x) \cdot 5}$

Step 5.3.7

Simplify the denominator.

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Step 5.3.7.1

Apply the distributive property.

$f (\frac{5 x + 6}{1 - x}) = \frac{11 x}{5 x + 6 + 1 \cdot 5 - x \cdot 5}$

Step 5.3.7.2

Multiply $5$ by $1$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{11 x}{5 x + 6 + 5 - x \cdot 5}$

Step 5.3.7.3

Multiply $5$ by $- 1$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{11 x}{5 x + 6 + 5 - 5 x}$

Step 5.3.7.4

Subtract $5 x$ from $5 x$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{11 x}{0 + 6 + 5}$

Step 5.3.7.5

Add $0$ and $6$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{11 x}{6 + 5}$

Step 5.3.7.6

Add $6$ and $5$ .

$f (\frac{5 x + 6}{1 - x}) = \frac{11 x}{11}$

Step 5.3.8

Cancel the common factor of $11$ .

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Step 5.3.8.1

Cancel the common factor.

$f (\frac{5 x + 6}{1 - x}) = \frac{11 x}{11}$

Step 5.3.8.2

Divide $x$ by $1$ .

$f (\frac{5 x + 6}{1 - x}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x + 6}{1 - x}$ is the inverse of $f (x) = \frac{x - 6}{x + 5}$ .

$f^{- 1} (x) = \frac{5 x + 6}{1 - x}$