Precalculus Examples

$f (x) = \frac{2 x + 5}{x - 7}$

Step 1

Write $f (x) = \frac{2 x + 5}{x - 7}$ as an equation.

$y = \frac{2 x + 5}{x - 7}$

Step 2

Interchange the variables.

$x = \frac{2 y + 5}{y - 7}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y - 7$ .

$(y - 7) (x) = (\frac{2 y + 5}{y - 7}) (y - 7)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$y x - 7 x = (\frac{2 y + 5}{y - 7}) (y - 7)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $y - 7$ .

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Step 3.3.1.1

Cancel the common factor.

$y x - 7 x = \frac{2 y + 5}{y - 7} (y - 7)$

Step 3.3.1.2

Rewrite the expression.

$y x - 7 x = 2 y + 5$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $2 y$ from both sides of the equation.

$y x - 7 x - 2 y = 5$

Step 3.4.2

Add $7 x$ to both sides of the equation.

$y x - 2 y = 5 + 7 x$

Step 3.4.3

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

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Step 3.4.3.1

Factor $y$ out of $y x$ .

$y (x) - 2 y = 5 + 7 x$

Step 3.4.3.2

Factor $y$ out of $- 2 y$ .

$y (x) + y \cdot - 2 = 5 + 7 x$

Step 3.4.3.3

Factor $y$ out of $y (x) + y \cdot - 2$ .

$y (x - 2) = 5 + 7 x$

Step 3.4.4

Divide each term in $y (x - 2) = 5 + 7 x$ by $x - 2$ and simplify.

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Step 3.4.4.1

Divide each term in $y (x - 2) = 5 + 7 x$ by $x - 2$ .

$\frac{y (x - 2)}{x - 2} = \frac{5}{x - 2} + \frac{7 x}{x - 2}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $x - 2$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (x - 2)}{x - 2} = \frac{5}{x - 2} + \frac{7 x}{x - 2}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{5}{x - 2} + \frac{7 x}{x - 2}$

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.

$y = \frac{5 + 7 x}{x - 2}$

Step 4

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

$f^{- 1} (x) = \frac{5 + 7 x}{x - 2}$

Step 5

Verify if $f^{- 1} (x) = \frac{5 + 7 x}{x - 2}$ is the inverse of $f (x) = \frac{2 x + 5}{x - 7}$ .

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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 $f^{- 1} (\frac{2 x + 5}{x - 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{5 + 7 (\frac{2 x + 5}{x - 7})}{(\frac{2 x + 5}{x - 7}) - 2}$

Step 5.2.3

Multiply the numerator and denominator of the fraction by $x - 7$ .

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Step 5.2.3.1

Multiply $\frac{5 + 7 \frac{2 x + 5}{x - 7}}{\frac{2 x + 5}{x - 7} - 2}$ by $\frac{x - 7}{x - 7}$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{x - 7}{x - 7} \cdot \frac{5 + 7 (\frac{2 x + 5}{x - 7})}{\frac{2 x + 5}{x - 7} - 2}$

Step 5.2.3.2

Combine.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (5 + 7 (\frac{2 x + 5}{x - 7}))}{(x - 7) (\frac{2 x + 5}{x - 7} - 2)}$

Step 5.2.4

Apply the distributive property.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) \cdot 5 + (x - 7) (7 (\frac{2 x + 5}{x - 7}))}{(x - 7) (\frac{2 x + 5}{x - 7}) + (x - 7) \cdot - 2}$

Step 5.2.5

Cancel the common factor of $x - 7$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) \cdot 5 + (x - 7) (7 (\frac{2 x + 5}{x - 7}))}{(x - 7) (\frac{2 x + 5}{x - 7}) + (x - 7) \cdot - 2}$

Step 5.2.5.2

Rewrite the expression.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) \cdot 5 + (x - 7) (7 (\frac{2 x + 5}{x - 7}))}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Factor $x - 7$ out of $(x - 7) \cdot 5 + (x - 7) \cdot 7 \frac{2 x + 5}{x - 7}$ .

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Step 5.2.6.1.1

Factor $x - 7$ out of $(x - 7) \cdot 5$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (5) + (x - 7) (7 (\frac{2 x + 5}{x - 7}))}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.1.2

Factor $x - 7$ out of $(x - 7) (5) + (x - 7) (7 \frac{2 x + 5}{x - 7})$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (5 + 7 (\frac{2 x + 5}{x - 7}))}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.2

Combine $7$ and $\frac{2 x + 5}{x - 7}$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (5 + \frac{7 (2 x + 5)}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{5 (x - 7)}{x - 7} + \frac{7 (2 x + 5)}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{5 (x - 7) + 7 (2 x + 5)}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.5

Reorder terms.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{7 (2 x + 5) + 5 (x - 7)}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.6

Rewrite $\frac{7 (2 x + 5) + 5 (x - 7)}{x - 7}$ in a factored form.

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Step 5.2.6.6.1

Apply the distributive property.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{7 (2 x) + 7 \cdot 5 + 5 (x - 7)}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.6.2

Multiply $2$ by $7$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{14 x + 7 \cdot 5 + 5 (x - 7)}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.6.3

Multiply $7$ by $5$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{14 x + 35 + 5 (x - 7)}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.6.4

Apply the distributive property.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{14 x + 35 + 5 x + 5 \cdot - 7}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.6.5

Multiply $5$ by $- 7$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{14 x + 35 + 5 x - 35}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.6.6

Add $14 x$ and $5 x$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{19 x + 35 - 35}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.6.7

Subtract $35$ from $35$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{19 x + 0}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.6.6.8

Add $19 x$ and $0$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{19 x}{x - 7})}{2 x + 5 + (x - 7) \cdot - 2}$

Step 5.2.7

Simplify the denominator.

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Step 5.2.7.1

Apply the distributive property.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{19 x}{x - 7})}{2 x + 5 + x \cdot - 2 - 7 \cdot - 2}$

Step 5.2.7.2

Move $- 2$ to the left of $x$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{19 x}{x - 7})}{2 x + 5 - 2 \cdot x - 7 \cdot - 2}$

Step 5.2.7.3

Multiply $- 7$ by $- 2$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{19 x}{x - 7})}{2 x + 5 - 2 \cdot x + 14}$

Step 5.2.7.4

Subtract $2 x$ from $2 x$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{19 x}{x - 7})}{0 + 5 + 14}$

Step 5.2.7.5

Add $0$ and $5$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{19 x}{x - 7})}{5 + 14}$

Step 5.2.7.6

Add $5$ and $14$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{(x - 7) (\frac{19 x}{x - 7})}{19}$

Step 5.2.8

Reduce the expression by cancelling the common factors.

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Step 5.2.8.1

Factor $\frac{19 x}{x - 7}$ out of $\frac{(x - 7) \frac{19 x}{x - 7}}{19}$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{19 x}{x - 7} \cdot \frac{x - 7}{19}$

Step 5.2.8.2

Cancel the common factor of $19$ .

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Step 5.2.8.2.1

Factor $19$ out of $19 x$ .

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{19 (x)}{x - 7} \cdot \frac{x - 7}{19}$

Step 5.2.8.2.2

Cancel the common factor.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{19 x}{x - 7} \cdot \frac{x - 7}{19}$

Step 5.2.8.2.3

Rewrite the expression.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{x}{x - 7} \cdot (x - 7)$

Step 5.2.8.3

Cancel the common factor of $x - 7$ .

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Step 5.2.8.3.1

Cancel the common factor.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = \frac{x}{x - 7} \cdot (x - 7)$

Step 5.2.8.3.2

Rewrite the expression.

$f^{- 1} (\frac{2 x + 5}{x - 7}) = x$

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 $f (\frac{5 + 7 x}{x - 2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{2 (\frac{5 + 7 x}{x - 2}) + 5}{(\frac{5 + 7 x}{x - 2}) - 7}$

Step 5.3.3

Multiply the numerator and denominator of the fraction by $x - 2$ .

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Step 5.3.3.1

Multiply $\frac{2 \frac{5 + 7 x}{x - 2} + 5}{\frac{5 + 7 x}{x - 2} - 7}$ by $\frac{x - 2}{x - 2}$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{x - 2}{x - 2} \cdot \frac{2 (\frac{5 + 7 x}{x - 2}) + 5}{\frac{5 + 7 x}{x - 2} - 7}$

Step 5.3.3.2

Combine.

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (2 (\frac{5 + 7 x}{x - 2}) + 5)}{(x - 2) (\frac{5 + 7 x}{x - 2} - 7)}$

Step 5.3.4

Apply the distributive property.

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (2 (\frac{5 + 7 x}{x - 2})) + (x - 2) \cdot 5}{(x - 2) (\frac{5 + 7 x}{x - 2}) + (x - 2) \cdot - 7}$

Step 5.3.5

Cancel the common factor of $x - 2$ .

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Step 5.3.5.1

Cancel the common factor.

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (2 (\frac{5 + 7 x}{x - 2})) + (x - 2) \cdot 5}{(x - 2) (\frac{5 + 7 x}{x - 2}) + (x - 2) \cdot - 7}$

Step 5.3.5.2

Rewrite the expression.

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (2 (\frac{5 + 7 x}{x - 2})) + (x - 2) \cdot 5}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Factor $x - 2$ out of $(x - 2) \cdot 2 \frac{5 + 7 x}{x - 2} + (x - 2) \cdot 5$ .

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Step 5.3.6.1.1

Factor $x - 2$ out of $(x - 2) \cdot 2 \frac{5 + 7 x}{x - 2}$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (2 (\frac{5 + 7 x}{x - 2})) + (x - 2) \cdot 5}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.1.2

Factor $x - 2$ out of $(x - 2) \cdot 5$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (2 (\frac{5 + 7 x}{x - 2})) + (x - 2) (5)}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.1.3

Factor $x - 2$ out of $(x - 2) (2 \frac{5 + 7 x}{x - 2}) + (x - 2) (5)$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (2 (\frac{5 + 7 x}{x - 2}) + 5)}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.2

Combine $2$ and $\frac{5 + 7 x}{x - 2}$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{2 (5 + 7 x)}{x - 2} + 5)}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{2 (5 + 7 x)}{x - 2} + \frac{5 (x - 2)}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.4

Combine the numerators over the common denominator.

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{2 (5 + 7 x) + 5 (x - 2)}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.5

Reorder terms.

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{5 (x - 2) + 2 (5 + 7 x)}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.6

Rewrite $\frac{5 (x - 2) + 2 (5 + 7 x)}{x - 2}$ in a factored form.

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Step 5.3.6.6.1

Apply the distributive property.

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{5 x + 5 \cdot - 2 + 2 (5 + 7 x)}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.6.2

Multiply $5$ by $- 2$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{5 x - 10 + 2 (5 + 7 x)}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.6.3

Apply the distributive property.

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{5 x - 10 + 2 \cdot 5 + 2 (7 x)}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.6.4

Multiply $2$ by $5$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{5 x - 10 + 10 + 2 (7 x)}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.6.5

Multiply $7$ by $2$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{5 x - 10 + 10 + 14 x}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.6.6

Add $5 x$ and $14 x$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{19 x - 10 + 10}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.6.7

Add $- 10$ and $10$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{19 x + 0}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.6.6.8

Add $19 x$ and $0$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{19 x}{x - 2})}{5 + 7 x + (x - 2) \cdot - 7}$

Step 5.3.7

Simplify the denominator.

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Step 5.3.7.1

Apply the distributive property.

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{19 x}{x - 2})}{5 + 7 x + x \cdot - 7 - 2 \cdot - 7}$

Step 5.3.7.2

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

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{19 x}{x - 2})}{5 + 7 x - 7 \cdot x - 2 \cdot - 7}$

Step 5.3.7.3

Multiply $- 2$ by $- 7$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{19 x}{x - 2})}{5 + 7 x - 7 \cdot x + 14}$

Step 5.3.7.4

Add $5$ and $14$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{19 x}{x - 2})}{7 x - 7 x + 19}$

Step 5.3.7.5

Subtract $7 x$ from $7 x$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{19 x}{x - 2})}{0 + 19}$

Step 5.3.7.6

Add $0$ and $19$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{(x - 2) (\frac{19 x}{x - 2})}{19}$

Step 5.3.8

Reduce the expression by cancelling the common factors.

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Step 5.3.8.1

Factor $\frac{19 x}{x - 2}$ out of $\frac{(x - 2) \frac{19 x}{x - 2}}{19}$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{19 x}{x - 2} \cdot \frac{x - 2}{19}$

Step 5.3.8.2

Cancel the common factor of $19$ .

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Step 5.3.8.2.1

Factor $19$ out of $19 x$ .

$f (\frac{5 + 7 x}{x - 2}) = \frac{19 (x)}{x - 2} \cdot \frac{x - 2}{19}$

Step 5.3.8.2.2

Cancel the common factor.

$f (\frac{5 + 7 x}{x - 2}) = \frac{19 x}{x - 2} \cdot \frac{x - 2}{19}$

Step 5.3.8.2.3

Rewrite the expression.

$f (\frac{5 + 7 x}{x - 2}) = \frac{x}{x - 2} \cdot (x - 2)$

Step 5.3.8.3

Cancel the common factor of $x - 2$ .

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Step 5.3.8.3.1

Cancel the common factor.

$f (\frac{5 + 7 x}{x - 2}) = \frac{x}{x - 2} \cdot (x - 2)$

Step 5.3.8.3.2

Rewrite the expression.

$f (\frac{5 + 7 x}{x - 2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 + 7 x}{x - 2}$ is the inverse of $f (x) = \frac{2 x + 5}{x - 7}$ .

$f^{- 1} (x) = \frac{5 + 7 x}{x - 2}$