Calculus Examples

$\frac{x + 2}{x - 6}$

Step 1

Interchange the variables.

$x = \frac{y + 2}{y - 6}$

Step 2

Solve for $y$ .

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Step 2.1

Multiply the equation by $y - 6$ .

$(y - 6) (x) = (\frac{y + 2}{y - 6}) (y - 6)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Apply the distributive property.

$y x - 6 x = (\frac{y + 2}{y - 6}) (y - 6)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $y - 6$ .

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Step 2.3.1.1

Cancel the common factor.

$y x - 6 x = \frac{y + 2}{y - 6} (y - 6)$

Step 2.3.1.2

Rewrite the expression.

$y x - 6 x = y + 2$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Subtract $y$ from both sides of the equation.

$y x - 6 x - y = 2$

Step 2.4.2

Add $6 x$ to both sides of the equation.

$y x - y = 2 + 6 x$

Step 2.4.3

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

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Step 2.4.3.1

Factor $y$ out of $y x$ .

$y (x) - y = 2 + 6 x$

Step 2.4.3.2

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = 2 + 6 x$

Step 2.4.3.3

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

$y (x - 1) = 2 + 6 x$

Step 2.4.4

Divide each term in $y (x - 1) = 2 + 6 x$ by $x - 1$ and simplify.

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Step 2.4.4.1

Divide each term in $y (x - 1) = 2 + 6 x$ by $x - 1$ .

$\frac{y (x - 1)}{x - 1} = \frac{2}{x - 1} + \frac{6 x}{x - 1}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $x - 1$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{y (x - 1)}{x - 1} = \frac{2}{x - 1} + \frac{6 x}{x - 1}$

Step 2.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{x - 1} + \frac{6 x}{x - 1}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Combine the numerators over the common denominator.

$y = \frac{2 + 6 x}{x - 1}$

Step 2.4.4.3.2

Factor $2$ out of $2 + 6 x$ .

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Step 2.4.4.3.2.1

Factor $2$ out of $2$ .

$y = \frac{2 (1) + 6 x}{x - 1}$

Step 2.4.4.3.2.2

Factor $2$ out of $6 x$ .

$y = \frac{2 (1) + 2 (3 x)}{x - 1}$

Step 2.4.4.3.2.3

Factor $2$ out of $2 (1) + 2 (3 x)$ .

$y = \frac{2 (1 + 3 x)}{x - 1}$

Step 3

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

$f^{- 1} (x) = \frac{2 (1 + 3 x)}{x - 1}$

Step 4

Verify if $f^{- 1} (x) = \frac{2 (1 + 3 x)}{x - 1}$ is the inverse of $f (x) = \frac{x + 2}{x - 6}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\frac{x + 2}{x - 6})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (1 + 3 (\frac{x + 2}{x - 6}))}{(\frac{x + 2}{x - 6}) - 1}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Combine $3$ and $\frac{x + 2}{x - 6}$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (1 + \frac{3 (x + 2)}{x - 6})}{\frac{x + 2}{x - 6} - 1}$

Step 4.2.3.2

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

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{x - 6}{x - 6} + \frac{3 (x + 2)}{x - 6})}{\frac{x + 2}{x - 6} - 1}$

Step 4.2.3.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{x - 6 + 3 (x + 2)}{x - 6})}{\frac{x + 2}{x - 6} - 1}$

Step 4.2.3.4

Reorder terms.

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{x + 3 (x + 2) - 6}{x - 6})}{\frac{x + 2}{x - 6} - 1}$

Step 4.2.3.5

Rewrite $\frac{x + 3 (x + 2) - 6}{x - 6}$ in a factored form.

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Step 4.2.3.5.1

Apply the distributive property.

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{x + 3 x + 3 \cdot 2 - 6}{x - 6})}{\frac{x + 2}{x - 6} - 1}$

Step 4.2.3.5.2

Multiply $3$ by $2$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{x + 3 x + 6 - 6}{x - 6})}{\frac{x + 2}{x - 6} - 1}$

Step 4.2.3.5.3

Add $x$ and $3 x$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x + 6 - 6}{x - 6})}{\frac{x + 2}{x - 6} - 1}$

Step 4.2.3.5.4

Subtract $6$ from $6$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x + 0}{x - 6})}{\frac{x + 2}{x - 6} - 1}$

Step 4.2.3.5.5

Add $4 x$ and $0$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x}{x - 6})}{\frac{x + 2}{x - 6} - 1}$

Step 4.2.4

Simplify the denominator.

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Step 4.2.4.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x}{x - 6})}{\frac{x + 2}{x - 6} - 1 \cdot \frac{x - 6}{x - 6}}$

Step 4.2.4.2

Combine $- 1$ and $\frac{x - 6}{x - 6}$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x}{x - 6})}{\frac{x + 2}{x - 6} + \frac{- (x - 6)}{x - 6}}$

Step 4.2.4.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x}{x - 6})}{\frac{x + 2 - (x - 6)}{x - 6}}$

Step 4.2.4.4

Rewrite $\frac{x + 2 - (x - 6)}{x - 6}$ in a factored form.

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Step 4.2.4.4.1

Apply the distributive property.

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x}{x - 6})}{\frac{x + 2 - x + 6}{x - 6}}$

Step 4.2.4.4.2

Multiply $- 1$ by $- 6$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x}{x - 6})}{\frac{x + 2 - x + 6}{x - 6}}$

Step 4.2.4.4.3

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x}{x - 6})}{\frac{0 + 2 + 6}{x - 6}}$

Step 4.2.4.4.4

Add $0$ and $2$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x}{x - 6})}{\frac{2 + 6}{x - 6}}$

Step 4.2.4.4.5

Add $2$ and $6$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{2 (\frac{4 x}{x - 6})}{\frac{8}{x - 6}}$

Step 4.2.5

Combine $2$ and $\frac{4 x}{x - 6}$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{\frac{2 (4 x)}{x - 6}}{\frac{8}{x - 6}}$

Step 4.2.6

Multiply $2$ by $4$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{\frac{8 x}{x - 6}}{\frac{8}{x - 6}}$

Step 4.2.7

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{8 x}{x - 6} \cdot \frac{x - 6}{8}$

Step 4.2.8

Cancel the common factor of $8$ .

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Step 4.2.8.1

Factor $8$ out of $8 x$ .

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{8 (x)}{x - 6} \cdot \frac{x - 6}{8}$

Step 4.2.8.2

Cancel the common factor.

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{8 x}{x - 6} \cdot \frac{x - 6}{8}$

Step 4.2.8.3

Rewrite the expression.

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{x}{x - 6} \cdot (x - 6)$

Step 4.2.9

Cancel the common factor of $x - 6$ .

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Step 4.2.9.1

Cancel the common factor.

$f^{- 1} (\frac{x + 2}{x - 6}) = \frac{x}{x - 6} \cdot (x - 6)$

Step 4.2.9.2

Rewrite the expression.

$f^{- 1} (\frac{x + 2}{x - 6}) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{2 (1 + 3 x)}{x - 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{(\frac{2 (1 + 3 x)}{x - 1}) + 2}{(\frac{2 (1 + 3 x)}{x - 1}) - 6}$

Step 4.3.3

Cancel the common factor of $(\frac{2 (1 + 3 x)}{x - 1}) + 2$ and $(\frac{2 (1 + 3 x)}{x - 1}) - 6$ .

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Step 4.3.3.1

Factor $2$ out of $\frac{2 (1 + 3 x)}{x - 1}$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{2 (\frac{1 + 3 x}{x - 1}) + 2}{(\frac{2 (1 + 3 x)}{x - 1}) - 6}$

Step 4.3.3.2

Factor $2$ out of $2$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{2 (\frac{1 + 3 x}{x - 1}) + 2 \cdot 1}{\frac{2 (1 + 3 x)}{x - 1} - 6}$

Step 4.3.3.3

Factor $2$ out of $2 \frac{1 + 3 x}{x - 1} + 2 (1)$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{2 (\frac{1 + 3 x}{x - 1} + 1)}{(\frac{2 (1 + 3 x)}{x - 1}) - 6}$

Step 4.3.3.4

Cancel the common factors.

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Step 4.3.3.4.1

Factor $2$ out of $\frac{2 (1 + 3 x)}{x - 1}$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{2 (\frac{1 + 3 x}{x - 1} + 1)}{2 (\frac{1 + 3 x}{x - 1}) - 6}$

Step 4.3.3.4.2

Factor $2$ out of $- 6$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{2 (\frac{1 + 3 x}{x - 1} + 1)}{2 (\frac{1 + 3 x}{x - 1}) + 2 \cdot - 3}$

Step 4.3.3.4.3

Factor $2$ out of $2 \frac{1 + 3 x}{x - 1} + 2 \cdot - 3$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{2 (\frac{1 + 3 x}{x - 1} + 1)}{2 (\frac{1 + 3 x}{x - 1} - 3)}$

Step 4.3.3.4.4

Cancel the common factor.

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{2 (\frac{1 + 3 x}{x - 1} + 1)}{2 (\frac{1 + 3 x}{x - 1} - 3)}$

Step 4.3.3.4.5

Rewrite the expression.

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{\frac{1 + 3 x}{x - 1} + 1}{\frac{1 + 3 x}{x - 1} - 3}$

Step 4.3.4

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

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Step 4.3.4.1

Multiply $\frac{\frac{1 + 3 x}{x - 1} + 1}{\frac{1 + 3 x}{x - 1} - 3}$ by $\frac{x - 1}{x - 1}$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{x - 1}{x - 1} \cdot \frac{\frac{1 + 3 x}{x - 1} + 1}{\frac{1 + 3 x}{x - 1} - 3}$

Step 4.3.4.2

Combine.

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{(x - 1) (\frac{1 + 3 x}{x - 1} + 1)}{(x - 1) (\frac{1 + 3 x}{x - 1} - 3)}$

Step 4.3.5

Apply the distributive property.

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{(x - 1) (\frac{1 + 3 x}{x - 1}) + (x - 1) \cdot 1}{(x - 1) (\frac{1 + 3 x}{x - 1}) + (x - 1) \cdot - 3}$

Step 4.3.6

Simplify by cancelling.

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Step 4.3.6.1

Cancel the common factor of $x - 1$ .

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Step 4.3.6.1.1

Cancel the common factor.

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{(x - 1) (\frac{1 + 3 x}{x - 1}) + (x - 1) \cdot 1}{(x - 1) (\frac{1 + 3 x}{x - 1}) + (x - 1) \cdot - 3}$

Step 4.3.6.1.2

Rewrite the expression.

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{1 + 3 x + (x - 1) \cdot 1}{(x - 1) (\frac{1 + 3 x}{x - 1}) + (x - 1) \cdot - 3}$

Step 4.3.6.2

Cancel the common factor of $x - 1$ .

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Step 4.3.6.2.1

Cancel the common factor.

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{1 + 3 x + (x - 1) \cdot 1}{(x - 1) (\frac{1 + 3 x}{x - 1}) + (x - 1) \cdot - 3}$

Step 4.3.6.2.2

Rewrite the expression.

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{1 + 3 x + (x - 1) \cdot 1}{1 + 3 x + (x - 1) \cdot - 3}$

Step 4.3.7

Simplify the numerator.

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Step 4.3.7.1

Multiply $x - 1$ by $1$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{1 + 3 x + x - 1}{1 + 3 x + (x - 1) \cdot - 3}$

Step 4.3.7.2

Subtract $1$ from $1$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{3 x + x + 0}{1 + 3 x + (x - 1) \cdot - 3}$

Step 4.3.7.3

Add $3 x + x$ and $0$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{3 x + x}{1 + 3 x + (x - 1) \cdot - 3}$

Step 4.3.7.4

Add $3 x$ and $x$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{4 x}{1 + 3 x + (x - 1) \cdot - 3}$

Step 4.3.8

Simplify the denominator.

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Step 4.3.8.1

Apply the distributive property.

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{4 x}{1 + 3 x + x \cdot - 3 - 1 \cdot - 3}$

Step 4.3.8.2

Move $- 3$ to the left of $x$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{4 x}{1 + 3 x - 3 \cdot x - 1 \cdot - 3}$

Step 4.3.8.3

Multiply $- 1$ by $- 3$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{4 x}{1 + 3 x - 3 \cdot x + 3}$

Step 4.3.8.4

Add $1$ and $3$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{4 x}{3 x - 3 x + 4}$

Step 4.3.8.5

Subtract $3 x$ from $3 x$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{4 x}{0 + 4}$

Step 4.3.8.6

Add $0$ and $4$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{4 x}{4}$

Step 4.3.9

Cancel the common factor of $4$ .

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Step 4.3.9.1

Cancel the common factor.

$f (\frac{2 (1 + 3 x)}{x - 1}) = \frac{4 x}{4}$

Step 4.3.9.2

Divide $x$ by $1$ .

$f (\frac{2 (1 + 3 x)}{x - 1}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 (1 + 3 x)}{x - 1}$ is the inverse of $f (x) = \frac{x + 2}{x - 6}$ .

$f^{- 1} (x) = \frac{2 (1 + 3 x)}{x - 1}$