Trigonometry Examples

$- \frac{x + 4}{3 x - 5}$

Step 1

Interchange the variables.

$x = - \frac{y + 4}{3 y - 5}$

Step 2

Solve for $y$ .

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Step 2.1

Multiply the equation by $3 y - 5$ .

$(3 y - 5) (x) = (- \frac{y + 4}{3 y - 5}) (3 y - 5)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Apply the distributive property.

$3 y x - 5 x = (- \frac{y + 4}{3 y - 5}) (3 y - 5)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify $(- \frac{y + 4}{3 y - 5}) (3 y - 5)$ .

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Step 2.3.1.1

Cancel the common factor of $3 y - 5$ .

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Step 2.3.1.1.1

Move the leading negative in $- \frac{y + 4}{3 y - 5}$ into the numerator.

$3 y x - 5 x = \frac{- (y + 4)}{3 y - 5} (3 y - 5)$

Step 2.3.1.1.2

Cancel the common factor.

$3 y x - 5 x = \frac{- (y + 4)}{3 y - 5} (3 y - 5)$

Step 2.3.1.1.3

Rewrite the expression.

$3 y x - 5 x = - (y + 4)$

Step 2.3.1.2

Apply the distributive property.

$3 y x - 5 x = - y - 1 \cdot 4$

Step 2.3.1.3

Multiply $- 1$ by $4$ .

$3 y x - 5 x = - y - 4$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Add $y$ to both sides of the equation.

$3 y x - 5 x + y = - 4$

Step 2.4.2

Add $5 x$ to both sides of the equation.

$3 y x + y = - 4 + 5 x$

Step 2.4.3

Factor $y$ out of $3 y x + y$ .

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Step 2.4.3.1

Factor $y$ out of $3 y x$ .

$y (3 x) + y = - 4 + 5 x$

Step 2.4.3.2

Raise $y$ to the power of $1$ .

$y (3 x) + y = - 4 + 5 x$

Step 2.4.3.3

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

$y (3 x) + y \cdot 1 = - 4 + 5 x$

Step 2.4.3.4

Factor $y$ out of $y (3 x) + y \cdot 1$ .

$y (3 x + 1) = - 4 + 5 x$

Step 2.4.4

Divide each term in $y (3 x + 1) = - 4 + 5 x$ by $3 x + 1$ and simplify.

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Step 2.4.4.1

Divide each term in $y (3 x + 1) = - 4 + 5 x$ by $3 x + 1$ .

$\frac{y (3 x + 1)}{3 x + 1} = \frac{- 4}{3 x + 1} + \frac{5 x}{3 x + 1}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $3 x + 1$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{y (3 x + 1)}{3 x + 1} = \frac{- 4}{3 x + 1} + \frac{5 x}{3 x + 1}$

Step 2.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4}{3 x + 1} + \frac{5 x}{3 x + 1}$

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.

$y = \frac{- 4 + 5 x}{3 x + 1}$

Step 2.4.4.3.2

Rewrite $- 4$ as $- 1 (4)$ .

$y = \frac{- 1 (4) + 5 x}{3 x + 1}$

Step 2.4.4.3.3

Factor $- 1$ out of $5 x$ .

$y = \frac{- 1 (4) - (- 5 x)}{3 x + 1}$

Step 2.4.4.3.4

Factor $- 1$ out of $- 1 (4) - (- 5 x)$ .

$y = \frac{- 1 (4 - 5 x)}{3 x + 1}$

Step 2.4.4.3.5

Move the negative in front of the fraction.

$y = - \frac{4 - 5 x}{3 x + 1}$

Step 3

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

$f^{- 1} (x) = - \frac{4 - 5 x}{3 x + 1}$

Step 4

Verify if $f^{- 1} (x) = - \frac{4 - 5 x}{3 x + 1}$ is the inverse of $f (x) = - \frac{x + 4}{3 x - 5}$ .

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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 $f^{- 1} (- \frac{x + 4}{3 x - 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{4 - 5 (- \frac{x + 4}{3 x - 5})}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Multiply $- 5 (- \frac{x + 4}{3 x - 5})$ .

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Step 4.2.3.1.1

Multiply $- 1$ by $- 5$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{4 + 5 (\frac{x + 4}{3 x - 5})}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.1.2

Combine $5$ and $\frac{x + 4}{3 x - 5}$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{4 + \frac{5 (x + 4)}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3 x - 5}{3 x - 5}$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{4 (3 x - 5)}{3 x - 5} + \frac{5 (x + 4)}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.3

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{4 (3 x - 5) + 5 (x + 4)}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.4

Reorder terms.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{5 (x + 4) + 4 (3 x - 5)}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.5

Rewrite $\frac{5 (x + 4) + 4 (3 x - 5)}{3 x - 5}$ in a factored form.

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Step 4.2.3.5.1

Apply the distributive property.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{5 x + 5 \cdot 4 + 4 (3 x - 5)}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.5.2

Multiply $5$ by $4$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{5 x + 20 + 4 (3 x - 5)}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.5.3

Apply the distributive property.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{5 x + 20 + 4 (3 x) + 4 \cdot - 5}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.5.4

Multiply $3$ by $4$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{5 x + 20 + 12 x + 4 \cdot - 5}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.5.5

Multiply $4$ by $- 5$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{5 x + 20 + 12 x - 20}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.5.6

Add $5 x$ and $12 x$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x + 20 - 20}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.5.7

Subtract $20$ from $20$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x + 0}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.3.5.8

Add $17 x$ and $0$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{3 (- \frac{x + 4}{3 x - 5}) + 1}$

Step 4.2.4

Simplify the denominator.

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Step 4.2.4.1

Multiply $3 (- \frac{x + 4}{3 x - 5})$ .

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Step 4.2.4.1.1

Multiply $- 1$ by $3$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{- 3 \frac{x + 4}{3 x - 5} + 1}$

Step 4.2.4.1.2

Combine $- 3$ and $\frac{x + 4}{3 x - 5}$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{\frac{- 3 (x + 4)}{3 x - 5} + 1}$

Step 4.2.4.2

Move the negative in front of the fraction.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{- \frac{3 (x + 4)}{3 x - 5} + 1}$

Step 4.2.4.3

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

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{- \frac{3 (x + 4)}{3 x - 5} + \frac{3 x - 5}{3 x - 5}}$

Step 4.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{\frac{- 3 (x + 4) + 3 x - 5}{3 x - 5}}$

Step 4.2.4.5

Reorder terms.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{\frac{3 x - 5 - 3 (x + 4)}{3 x - 5}}$

Step 4.2.4.6

Rewrite $\frac{3 x - 5 - 3 (x + 4)}{3 x - 5}$ in a factored form.

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Step 4.2.4.6.1

Apply the distributive property.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{\frac{3 x - 5 - 3 x - 3 \cdot 4}{3 x - 5}}$

Step 4.2.4.6.2

Multiply $- 3$ by $4$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{\frac{3 x - 5 - 3 x - 12}{3 x - 5}}$

Step 4.2.4.6.3

Subtract $3 x$ from $3 x$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{\frac{0 - 5 - 12}{3 x - 5}}$

Step 4.2.4.6.4

Subtract $5$ from $0$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{\frac{- 5 - 12}{3 x - 5}}$

Step 4.2.4.6.5

Subtract $12$ from $- 5$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{\frac{- 17}{3 x - 5}}$

Step 4.2.4.7

Move the negative in front of the fraction.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - \frac{\frac{17 x}{3 x - 5}}{- \frac{17}{3 x - 5}}$

Step 4.2.5

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - (\frac{17 x}{3 x - 5} \cdot (- \frac{3 x - 5}{17}))$

Step 4.2.6

Rewrite using the commutative property of multiplication.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - (- \frac{17 x}{3 x - 5} \cdot \frac{3 x - 5}{17})$

Step 4.2.7

Cancel the common factor of $17$ .

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Step 4.2.7.1

Move the leading negative in $- \frac{17 x}{3 x - 5}$ into the numerator.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - (\frac{- 17 x}{3 x - 5} \cdot \frac{3 x - 5}{17})$

Step 4.2.7.2

Factor $17$ out of $- 17 x$ .

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - (\frac{17 (- x)}{3 x - 5} \cdot \frac{3 x - 5}{17})$

Step 4.2.7.3

Cancel the common factor.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - (\frac{17 (- x)}{3 x - 5} \cdot \frac{3 x - 5}{17})$

Step 4.2.7.4

Rewrite the expression.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - (\frac{- x}{3 x - 5} \cdot (3 x - 5))$

Step 4.2.8

Cancel the common factor of $3 x - 5$ .

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Step 4.2.8.1

Cancel the common factor.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = - (\frac{- x}{3 x - 5} \cdot (3 x - 5))$

Step 4.2.8.2

Rewrite the expression.

$f^{- 1} (- \frac{x + 4}{3 x - 5}) = x$

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 $f (- \frac{4 - 5 x}{3 x + 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{(- \frac{4 - 5 x}{3 x + 1}) + 4}{3 (- \frac{4 - 5 x}{3 x + 1}) - 5}$

Step 4.3.3

Multiply the numerator and denominator of the fraction by $3 x + 1$ .

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Step 4.3.3.1

Multiply $\frac{- \frac{4 - 5 x}{3 x + 1} + 4}{3 (- \frac{4 - 5 x}{3 x + 1}) - 5}$ by $\frac{3 x + 1}{3 x + 1}$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - (\frac{3 x + 1}{3 x + 1} \cdot \frac{- \frac{4 - 5 x}{3 x + 1} + 4}{3 (- \frac{4 - 5 x}{3 x + 1}) - 5})$

Step 4.3.3.2

Combine.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{(3 x + 1) (- \frac{4 - 5 x}{3 x + 1} + 4)}{(3 x + 1) (3 (- \frac{4 - 5 x}{3 x + 1}) - 5)}$

Step 4.3.4

Apply the distributive property.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{(3 x + 1) (- \frac{4 - 5 x}{3 x + 1}) + (3 x + 1) \cdot 4}{(3 x + 1) (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.5

Cancel the common factor of $3 x + 1$ .

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Step 4.3.5.1

Move the leading negative in $- \frac{4 - 5 x}{3 x + 1}$ into the numerator.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{(3 x + 1) (\frac{- (4 - 5 x)}{3 x + 1}) + (3 x + 1) \cdot 4}{(3 x + 1) (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.5.2

Cancel the common factor.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{(3 x + 1) (\frac{- (4 - 5 x)}{3 x + 1}) + (3 x + 1) \cdot 4}{(3 x + 1) (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.5.3

Rewrite the expression.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{- (4 - 5 x) + (3 x + 1) \cdot 4}{(3 x + 1) (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.6

Simplify the numerator.

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Step 4.3.6.1

Apply the distributive property.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{- 1 \cdot 4 - (- 5 x) + (3 x + 1) \cdot 4}{(3 x + 1) \cdot (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.6.2

Multiply $- 1$ by $4$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{- 4 - (- 5 x) + (3 x + 1) \cdot 4}{(3 x + 1) \cdot (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.6.3

Multiply $- 5$ by $- 1$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{- 4 + 5 x + (3 x + 1) \cdot 4}{(3 x + 1) \cdot (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.6.4

Apply the distributive property.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{- 4 + 5 x + 3 x \cdot 4 + 1 \cdot 4}{(3 x + 1) \cdot (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.6.5

Multiply $4$ by $3$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{- 4 + 5 x + 12 x + 1 \cdot 4}{(3 x + 1) \cdot (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.6.6

Multiply $4$ by $1$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{- 4 + 5 x + 12 x + 4}{(3 x + 1) \cdot (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.6.7

Add $- 4$ and $4$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{5 x + 12 x + 0}{(3 x + 1) \cdot (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.6.8

Add $5 x + 12 x$ and $0$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{5 x + 12 x}{(3 x + 1) \cdot (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.6.9

Add $5 x$ and $12 x$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) \cdot (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) \cdot - 5}$

Step 4.3.7

Simplify the denominator.

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Step 4.3.7.1

Factor $3 x + 1$ out of $(3 x + 1) \cdot 3 (- \frac{4 - 5 x}{3 x + 1}) + (3 x + 1) \cdot - 5$ .

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Step 4.3.7.1.1

Factor $3 x + 1$ out of $(3 x + 1) \cdot - 5$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) (- 5)}$

Step 4.3.7.1.2

Factor $3 x + 1$ out of $(3 x + 1) (3 (- \frac{4 - 5 x}{3 x + 1})) + (3 x + 1) (- 5)$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (3 (- \frac{4 - 5 x}{3 x + 1}) - 5)}$

Step 4.3.7.2

Multiply $3 (- \frac{4 - 5 x}{3 x + 1})$ .

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Step 4.3.7.2.1

Multiply $- 1$ by $3$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (- 3 \frac{4 - 5 x}{3 x + 1} - 5)}$

Step 4.3.7.2.2

Combine $- 3$ and $\frac{4 - 5 x}{3 x + 1}$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- 3 (4 - 5 x)}{3 x + 1} - 5)}$

Step 4.3.7.3

Move the negative in front of the fraction.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (- \frac{3 (4 - 5 x)}{3 x + 1} - 5)}$

Step 4.3.7.4

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3 x + 1}{3 x + 1}$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (- \frac{3 (4 - 5 x)}{3 x + 1} - 5 \cdot \frac{3 x + 1}{3 x + 1})}$

Step 4.3.7.5

Combine $- 5$ and $\frac{3 x + 1}{3 x + 1}$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (- \frac{3 (4 - 5 x)}{3 x + 1} + \frac{- 5 (3 x + 1)}{3 x + 1})}$

Step 4.3.7.6

Combine the numerators over the common denominator.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- 3 (4 - 5 x) - 5 (3 x + 1)}{3 x + 1})}$

Step 4.3.7.7

Rewrite $\frac{- 3 (4 - 5 x) - 5 (3 x + 1)}{3 x + 1}$ in a factored form.

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Step 4.3.7.7.1

Factor $- 1$ out of $- 3 (4 - 5 x) - 5 (3 x + 1)$ .

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Step 4.3.7.7.1.1

Reorder $4$ and $- 5 x$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- 3 (- 5 x + 4) - 5 (3 x + 1)}{3 x + 1})}$

Step 4.3.7.7.1.2

Factor $- 1$ out of $- 3 (- 5 x + 4)$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (3 (- 5 x + 4)) - 5 (3 x + 1)}{3 x + 1})}$

Step 4.3.7.7.1.3

Factor $- 1$ out of $- 5 (3 x + 1)$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (3 (- 5 x + 4)) - (5 (3 x + 1))}{3 x + 1})}$

Step 4.3.7.7.1.4

Factor $- 1$ out of $- (3 (- 5 x + 4)) - (5 (3 x + 1))$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (3 (- 5 x + 4) + 5 (3 x + 1))}{3 x + 1})}$

Step 4.3.7.7.2

Apply the distributive property.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (3 (- 5 x) + 3 \cdot 4 + 5 (3 x + 1))}{3 x + 1})}$

Step 4.3.7.7.3

Multiply $- 5$ by $3$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (- 15 x + 3 \cdot 4 + 5 (3 x + 1))}{3 x + 1})}$

Step 4.3.7.7.4

Multiply $3$ by $4$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (- 15 x + 12 + 5 (3 x + 1))}{3 x + 1})}$

Step 4.3.7.7.5

Apply the distributive property.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (- 15 x + 12 + 5 (3 x) + 5 \cdot 1)}{3 x + 1})}$

Step 4.3.7.7.6

Multiply $3$ by $5$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (- 15 x + 12 + 15 x + 5 \cdot 1)}{3 x + 1})}$

Step 4.3.7.7.7

Multiply $5$ by $1$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (- 15 x + 12 + 15 x + 5)}{3 x + 1})}$

Step 4.3.7.7.8

Add $- 15 x$ and $15 x$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (0 + 12 + 5)}{3 x + 1})}$

Step 4.3.7.7.9

Add $0$ and $12$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- (12 + 5)}{3 x + 1})}$

Step 4.3.7.7.10

Add $12$ and $5$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- 1 \cdot 17}{3 x + 1})}$

Step 4.3.7.8

Multiply $- 1$ by $17$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) (\frac{- 17}{3 x + 1})}$

Step 4.3.7.9

Move the negative in front of the fraction.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{(3 x + 1) \cdot (- 1 \frac{17}{3 x + 1})}$

Step 4.3.7.10

Factor out negative.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{17 x}{- (3 x + 1) \frac{17}{3 x + 1}}$

Step 4.3.8

Factor $\frac{17}{3 x + 1}$ out of $\frac{17 x}{- (3 x + 1) \frac{17}{3 x + 1}}$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - (\frac{3 x + 1}{17} \cdot \frac{17 x}{- (3 x + 1)})$

Step 4.3.9

Cancel the common factor of $3 x + 1$ .

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Step 4.3.9.1

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

$f (- \frac{4 - 5 x}{3 x + 1}) = - (\frac{3 x + 1}{17} \cdot \frac{17 x}{(3 x + 1) \cdot - 1})$

Step 4.3.9.2

Cancel the common factor.

$f (- \frac{4 - 5 x}{3 x + 1}) = - (\frac{3 x + 1}{17} \cdot \frac{17 x}{(3 x + 1) \cdot - 1})$

Step 4.3.9.3

Rewrite the expression.

$f (- \frac{4 - 5 x}{3 x + 1}) = - (\frac{1}{17} \cdot \frac{17 x}{- 1})$

Step 4.3.10

Cancel the common factor of $17$ .

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Step 4.3.10.1

Factor $17$ out of $17 x$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - (\frac{1}{17} \cdot \frac{17 (x)}{- 1})$

Step 4.3.10.2

Cancel the common factor.

$f (- \frac{4 - 5 x}{3 x + 1}) = - (\frac{1}{17} \cdot \frac{17 x}{- 1})$

Step 4.3.10.3

Rewrite the expression.

$f (- \frac{4 - 5 x}{3 x + 1}) = - \frac{x}{- 1}$

Step 4.3.11

Simplify the expression.

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Step 4.3.11.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = - (- 1 \cdot x)$

Step 4.3.11.2

Rewrite $- 1 \cdot x$ as $- x$ .

$f (- \frac{4 - 5 x}{3 x + 1}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{4 - 5 x}{3 x + 1}$ is the inverse of $f (x) = - \frac{x + 4}{3 x - 5}$ .

$f^{- 1} (x) = - \frac{4 - 5 x}{3 x + 1}$