Trigonometry Examples

$f (x) = \frac{- 3 x - 6}{5 x - 5}$

Step 1

Write $f (x) = \frac{- 3 x - 6}{5 x - 5}$ as an equation.

$y = \frac{- 3 x - 6}{5 x - 5}$

Step 2

Interchange the variables.

$x = \frac{- 3 y - 6}{5 y - 5}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{- 3 y - 6}{5 y - 5} = x$ .

$\frac{- 3 y - 6}{5 y - 5} = x$

Step 3.2

Factor each term.

Tap for more steps...

Step 3.2.1

Factor $3$ out of $- 3 y - 6$ .

Tap for more steps...

Step 3.2.1.1

Factor $3$ out of $- 3 y$ .

$\frac{3 (- y) - 6}{5 y - 5} = x$

Step 3.2.1.2

Factor $3$ out of $- 6$ .

$\frac{3 (- y) + 3 (- 2)}{5 y - 5} = x$

Step 3.2.1.3

Factor $3$ out of $3 (- y) + 3 (- 2)$ .

$\frac{3 (- y - 2)}{5 y - 5} = x$

Step 3.2.2

Factor $5$ out of $5 y - 5$ .

Tap for more steps...

Step 3.2.2.1

Factor $5$ out of $5 y$ .

$\frac{3 (- y - 2)}{5 (y) - 5} = x$

Step 3.2.2.2

Factor $5$ out of $- 5$ .

$\frac{3 (- y - 2)}{5 (y) + 5 (- 1)} = x$

Step 3.2.2.3

Factor $5$ out of $5 (y) + 5 (- 1)$ .

$\frac{3 (- y - 2)}{5 (y - 1)} = x$

Step 3.3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$5 (y - 1), 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$5 (y - 1)$

Step 3.4

Multiply each term in $\frac{3 (- y - 2)}{5 (y - 1)} = x$ by $5 (y - 1)$ to eliminate the fractions.

Tap for more steps...

Step 3.4.1

Multiply each term in $\frac{3 (- y - 2)}{5 (y - 1)} = x$ by $5 (y - 1)$ .

$\frac{3 (- y - 2)}{5 (y - 1)} (5 (y - 1)) = x (5 (y - 1))$

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Rewrite using the commutative property of multiplication.

$5 \frac{3 (- y - 2)}{5 (y - 1)} (y - 1) = x (5 (y - 1))$

Step 3.4.2.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.4.2.2.1

Cancel the common factor.

$5 \frac{3 (- y - 2)}{5 (y - 1)} (y - 1) = x (5 (y - 1))$

Step 3.4.2.2.2

Rewrite the expression.

$\frac{3 (- y - 2)}{y - 1} (y - 1) = x (5 (y - 1))$

Step 3.4.2.3

Cancel the common factor of $y - 1$ .

Tap for more steps...

Step 3.4.2.3.1

Cancel the common factor.

$\frac{3 (- y - 2)}{y - 1} (y - 1) = x (5 (y - 1))$

Step 3.4.2.3.2

Rewrite the expression.

$3 (- y - 2) = x (5 (y - 1))$

Step 3.4.2.4

Apply the distributive property.

$3 (- y) + 3 \cdot - 2 = x (5 (y - 1))$

Step 3.4.2.5

Multiply.

Tap for more steps...

Step 3.4.2.5.1

Multiply $- 1$ by $3$ .

$- 3 y + 3 \cdot - 2 = x (5 (y - 1))$

Step 3.4.2.5.2

Multiply $3$ by $- 2$ .

$- 3 y - 6 = x (5 (y - 1))$

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Rewrite using the commutative property of multiplication.

$- 3 y - 6 = 5 x (y - 1)$

Step 3.4.3.2

Apply the distributive property.

$- 3 y - 6 = 5 x y + 5 x \cdot - 1$

Step 3.4.3.3

Multiply $- 1$ by $5$ .

$- 3 y - 6 = 5 x y - 5 x$

Step 3.5

Solve the equation.

Tap for more steps...

Step 3.5.1

Subtract $5 x y$ from both sides of the equation.

$- 3 y - 6 - 5 x y = - 5 x$

Step 3.5.2

Add $6$ to both sides of the equation.

$- 3 y - 5 x y = - 5 x + 6$

Step 3.5.3

Factor $y$ out of $- 3 y - 5 x y$ .

Tap for more steps...

Step 3.5.3.1

Factor $y$ out of $- 3 y$ .

$y \cdot - 3 - 5 x y = - 5 x + 6$

Step 3.5.3.2

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

$y \cdot - 3 + y (- 5 x) = - 5 x + 6$

Step 3.5.3.3

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

$y (- 3 - 5 x) = - 5 x + 6$

Step 3.5.4

Divide each term in $y (- 3 - 5 x) = - 5 x + 6$ by $- 3 - 5 x$ and simplify.

Tap for more steps...

Step 3.5.4.1

Divide each term in $y (- 3 - 5 x) = - 5 x + 6$ by $- 3 - 5 x$ .

$\frac{y (- 3 - 5 x)}{- 3 - 5 x} = \frac{- 5 x}{- 3 - 5 x} + \frac{6}{- 3 - 5 x}$

Step 3.5.4.2

Simplify the left side.

Tap for more steps...

Step 3.5.4.2.1

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

Tap for more steps...

Step 3.5.4.2.1.1

Cancel the common factor.

$\frac{y (- 3 - 5 x)}{- 3 - 5 x} = \frac{- 5 x}{- 3 - 5 x} + \frac{6}{- 3 - 5 x}$

Step 3.5.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5 x}{- 3 - 5 x} + \frac{6}{- 3 - 5 x}$

Step 3.5.4.3

Simplify the right side.

Tap for more steps...

Step 3.5.4.3.1

Combine the numerators over the common denominator.

$y = \frac{- 5 x + 6}{- 3 - 5 x}$

Step 3.5.4.3.2

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

$y = \frac{- (5 x) + 6}{- 3 - 5 x}$

Step 3.5.4.3.3

Rewrite $6$ as $- 1 (- 6)$ .

$y = \frac{- (5 x) - 1 (- 6)}{- 3 - 5 x}$

Step 3.5.4.3.4

Factor $- 1$ out of $- (5 x) - 1 (- 6)$ .

$y = \frac{- (5 x - 6)}{- 3 - 5 x}$

Step 3.5.4.3.5

Rewrite $- (5 x - 6)$ as $- 1 (5 x - 6)$ .

$y = \frac{- 1 (5 x - 6)}{- 3 - 5 x}$

Step 3.5.4.3.6

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{- 1 (5 x - 6)}{- 1 (3) - 5 x}$

Step 3.5.4.3.7

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

$y = \frac{- 1 (5 x - 6)}{- 1 (3) - (5 x)}$

Step 3.5.4.3.8

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

$y = \frac{- 1 (5 x - 6)}{- 1 (3 + 5 x)}$

Step 3.5.4.3.9

Cancel the common factor.

$y = \frac{- 1 (5 x - 6)}{- 1 (3 + 5 x)}$

Step 3.5.4.3.10

Rewrite the expression.

$y = \frac{5 x - 6}{3 + 5 x}$

Step 4

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

$f^{- 1} (x) = \frac{5 x - 6}{3 + 5 x}$

Step 5

Verify if $f^{- 1} (x) = \frac{5 x - 6}{3 + 5 x}$ is the inverse of $f (x) = \frac{- 3 x - 6}{5 x - 5}$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{- 3 x - 6}{5 x - 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{- 3 x - 6}{5 x - 5}) - 6}{3 + 5 (\frac{- 3 x - 6}{5 x - 5})}$

Step 5.2.3

Simplify with factoring out.

Tap for more steps...

Step 5.2.3.1

Factor $3$ out of $- 3 x - 6$ .

Tap for more steps...

Step 5.2.3.1.1

Factor $3$ out of $- 3 x$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x) - 6}{5 x - 5}) - 6}{3 + 5 (\frac{- 3 x - 6}{5 x - 5})}$

Step 5.2.3.1.2

Factor $3$ out of $- 6$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x) + 3 (- 2)}{5 x - 5}) - 6}{3 + 5 (\frac{- 3 x - 6}{5 x - 5})}$

Step 5.2.3.1.3

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

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 x - 5}) - 6}{3 + 5 (\frac{- 3 x - 6}{5 x - 5})}$

Step 5.2.3.2

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

Tap for more steps...

Step 5.2.3.2.1

Factor $5$ out of $5 x$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 (x) - 5}) - 6}{3 + 5 (\frac{- 3 x - 6}{5 x - 5})}$

Step 5.2.3.2.2

Factor $5$ out of $- 5$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 (x) + 5 (- 1)}) - 6}{3 + 5 (\frac{- 3 x - 6}{5 x - 5})}$

Step 5.2.3.2.3

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

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 (x - 1)}) - 6}{3 + 5 (\frac{- 3 x - 6}{5 x - 5})}$

Step 5.2.3.3

Factor $3$ out of $- 3 x - 6$ .

Tap for more steps...

Step 5.2.3.3.1

Factor $3$ out of $- 3 x$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 (x - 1)}) - 6}{3 + 5 (\frac{3 (- x) - 6}{5 x - 5})}$

Step 5.2.3.3.2

Factor $3$ out of $- 6$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 (x - 1)}) - 6}{3 + 5 (\frac{3 (- x) + 3 (- 2)}{5 x - 5})}$

Step 5.2.3.3.3

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

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 (x - 1)}) - 6}{3 + 5 (\frac{3 (- x - 2)}{5 x - 5})}$

Step 5.2.3.4

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

Tap for more steps...

Step 5.2.3.4.1

Factor $5$ out of $5 x$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 (x - 1)}) - 6}{3 + 5 (\frac{3 (- x - 2)}{5 (x) - 5})}$

Step 5.2.3.4.2

Factor $5$ out of $- 5$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 (x - 1)}) - 6}{3 + 5 (\frac{3 (- x - 2)}{5 (x) + 5 (- 1)})}$

Step 5.2.3.4.3

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

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 (x - 1)}) - 6}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4

Simplify the numerator.

Tap for more steps...

Step 5.2.4.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.4.1.1

Cancel the common factor.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{5 (\frac{3 (- x - 2)}{5 (x - 1)}) - 6}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.1.2

Rewrite the expression.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 (- x - 2)}{x - 1} - 6}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.2

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

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 (- x - 2)}{x - 1} - 6 \cdot \frac{x - 1}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.3

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

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 (- x - 2)}{x - 1} + \frac{- 6 (x - 1)}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 (- x - 2) - 6 (x - 1)}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.5

Rewrite $\frac{3 (- x - 2) - 6 (x - 1)}{x - 1}$ in a factored form.

Tap for more steps...

Step 5.2.4.5.1

Factor $3$ out of $3 (- x - 2) - 6 (x - 1)$ .

Tap for more steps...

Step 5.2.4.5.1.1

Factor $3$ out of $- 6 (x - 1)$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 (- x - 2) + 3 (- 2 (x - 1))}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.5.1.2

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

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 (- x - 2 - 2 (x - 1))}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.5.2

Apply the distributive property.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 (- x - 2 - 2 x - 2 \cdot - 1)}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.5.3

Multiply $- 2$ by $- 1$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 (- x - 2 - 2 x + 2)}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.5.4

Subtract $2 x$ from $- x$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 (- 3 x - 2 + 2)}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.5.5

Add $- 2$ and $2$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 (- 3 x + 0)}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.5.6

Add $- 3 x$ and $0$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{3 \cdot (- 3 x)}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.5.7

Multiply $3$ by $- 3$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{- 9 x}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.4.6

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.5

Simplify the denominator.

Tap for more steps...

Step 5.2.5.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.5.1.1

Cancel the common factor.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{3 + 5 (\frac{3 (- x - 2)}{5 (x - 1)})}$

Step 5.2.5.1.2

Rewrite the expression.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{3 + \frac{3 (- x - 2)}{x - 1}}$

Step 5.2.5.2

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

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{\frac{3 (x - 1)}{x - 1} + \frac{3 (- x - 2)}{x - 1}}$

Step 5.2.5.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{\frac{3 (x - 1) + 3 (- x - 2)}{x - 1}}$

Step 5.2.5.4

Rewrite $\frac{3 (x - 1) + 3 (- x - 2)}{x - 1}$ in a factored form.

Tap for more steps...

Step 5.2.5.4.1

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

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{\frac{3 (x - 1 - x - 2)}{x - 1}}$

Step 5.2.5.4.2

Subtract $x$ from $x$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{\frac{3 (0 - 1 - 2)}{x - 1}}$

Step 5.2.5.4.3

Subtract $1$ from $0$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{\frac{3 (- 1 - 2)}{x - 1}}$

Step 5.2.5.4.4

Subtract $2$ from $- 1$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{\frac{3 \cdot - 3}{x - 1}}$

Step 5.2.5.5

Multiply $3$ by $- 3$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{\frac{- 9}{x - 1}}$

Step 5.2.5.6

Move the negative in front of the fraction.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{- \frac{9 x}{x - 1}}{- \frac{9}{x - 1}}$

Step 5.2.6

Dividing two negative values results in a positive value.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{\frac{9 x}{x - 1}}{\frac{9}{x - 1}}$

Step 5.2.7

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{9 x}{x - 1} \cdot \frac{x - 1}{9}$

Step 5.2.8

Cancel the common factor of $9$ .

Tap for more steps...

Step 5.2.8.1

Factor $9$ out of $9 x$ .

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{9 (x)}{x - 1} \cdot \frac{x - 1}{9}$

Step 5.2.8.2

Cancel the common factor.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{9 x}{x - 1} \cdot \frac{x - 1}{9}$

Step 5.2.8.3

Rewrite the expression.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{x}{x - 1} \cdot (x - 1)$

Step 5.2.9

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 5.2.9.1

Cancel the common factor.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = \frac{x}{x - 1} \cdot (x - 1)$

Step 5.2.9.2

Rewrite the expression.

$f^{- 1} (\frac{- 3 x - 6}{5 x - 5}) = x$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{5 x - 6}{3 + 5 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{5 x - 6}{3 + 5 x} - 6}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3

Simplify the numerator.

Tap for more steps...

Step 5.3.3.1

Factor $- 3$ out of $- 3 \frac{5 x - 6}{3 + 5 x} - 6$ .

Tap for more steps...

Step 5.3.3.1.1

Reorder $3$ and $5 x$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{5 x - 6}{5 x + 3} - 6}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.1.2

Factor $- 3$ out of $- 6$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{5 x - 6}{5 x + 3} - 3 \cdot 2}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.1.3

Factor $- 3$ out of $- 3 (\frac{5 x - 6}{5 x + 3}) - 3 (2)$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 (\frac{5 x - 6}{5 x + 3} + 2)}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{5 x + 3}{5 x + 3}$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 (\frac{5 x - 6}{5 x + 3} + \frac{2 (5 x + 3)}{5 x + 3})}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.3

Combine the numerators over the common denominator.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{5 x - 6 + 2 (5 x + 3)}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.4

Reorder terms.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{5 x + 2 (5 x + 3) - 6}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.5

Rewrite $\frac{5 x + 2 (5 x + 3) - 6}{5 x + 3}$ in a factored form.

Tap for more steps...

Step 5.3.3.5.1

Apply the distributive property.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{5 x + 2 (5 x) + 2 \cdot 3 - 6}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.5.2

Multiply $5$ by $2$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{5 x + 10 x + 2 \cdot 3 - 6}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.5.3

Multiply $2$ by $3$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{5 x + 10 x + 6 - 6}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.5.4

Add $5 x$ and $10 x$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x + 6 - 6}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.5.5

Subtract $6$ from $6$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x + 0}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.3.5.6

Add $15 x$ and $0$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.4

Simplify the denominator.

Tap for more steps...

Step 5.3.4.1

Factor $5$ out of $5 \frac{5 x - 6}{3 + 5 x} - 5$ .

Tap for more steps...

Step 5.3.4.1.1

Factor $5$ out of $5 \frac{5 x - 6}{3 + 5 x}$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x}) - 5}$

Step 5.3.4.1.2

Factor $5$ out of $- 5$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x}) + 5 (- 1)}$

Step 5.3.4.1.3

Factor $5$ out of $5 (\frac{5 x - 6}{3 + 5 x}) + 5 (- 1)$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x} - 1)}$

Step 5.3.4.2

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

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x} - 1 \cdot \frac{3 + 5 x}{3 + 5 x})}$

Step 5.3.4.3

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

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6}{3 + 5 x} + \frac{- (3 + 5 x)}{3 + 5 x})}$

Step 5.3.4.4

Combine the numerators over the common denominator.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6 - (3 + 5 x)}{3 + 5 x})}$

Step 5.3.4.5

Reorder terms.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6 - (3 + 5 x)}{5 x + 3})}$

Step 5.3.4.6

Rewrite $\frac{5 x - 6 - (3 + 5 x)}{5 x + 3}$ in a factored form.

Tap for more steps...

Step 5.3.4.6.1

Apply the distributive property.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6 - 1 \cdot 3 - (5 x)}{5 x + 3})}$

Step 5.3.4.6.2

Multiply $- 1$ by $3$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6 - 3 - (5 x)}{5 x + 3})}$

Step 5.3.4.6.3

Multiply $5$ by $- 1$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{5 x - 6 - 3 - 5 x}{5 x + 3})}$

Step 5.3.4.6.4

Subtract $5 x$ from $5 x$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{0 - 6 - 3}{5 x + 3})}$

Step 5.3.4.6.5

Subtract $6$ from $0$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{- 6 - 3}{5 x + 3})}$

Step 5.3.4.6.6

Subtract $3$ from $- 6$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 (\frac{- 9}{5 x + 3})}$

Step 5.3.4.7

Move the negative in front of the fraction.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{5 \cdot (- 1 \frac{9}{5 x + 3})}$

Step 5.3.4.8

Combine exponents.

Tap for more steps...

Step 5.3.4.8.1

Factor out negative.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{- (5 (\frac{9}{5 x + 3}))}$

Step 5.3.4.8.2

Combine $5$ and $\frac{9}{5 x + 3}$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{- \frac{5 \cdot 9}{5 x + 3}}$

Step 5.3.4.8.3

Multiply $5$ by $9$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- 3 \frac{15 x}{5 x + 3}}{- \frac{45}{5 x + 3}}$

Step 5.3.5

Combine $- 3$ and $\frac{15 x}{5 x + 3}$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{\frac{- 3 (15 x)}{5 x + 3}}{- \frac{45}{5 x + 3}}$

Step 5.3.6

Multiply $- 3$ by $15$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{\frac{- 45 x}{5 x + 3}}{- \frac{45}{5 x + 3}}$

Step 5.3.7

Move the negative in front of the fraction.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{- \frac{45 x}{5 x + 3}}{- \frac{45}{5 x + 3}}$

Step 5.3.8

Dividing two negative values results in a positive value.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{\frac{45 x}{5 x + 3}}{\frac{45}{5 x + 3}}$

Step 5.3.9

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{45 x}{5 x + 3} \cdot \frac{5 x + 3}{45}$

Step 5.3.10

Cancel the common factor of $45$ .

Tap for more steps...

Step 5.3.10.1

Factor $45$ out of $45 x$ .

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{45 (x)}{5 x + 3} \cdot \frac{5 x + 3}{45}$

Step 5.3.10.2

Cancel the common factor.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{45 x}{5 x + 3} \cdot \frac{5 x + 3}{45}$

Step 5.3.10.3

Rewrite the expression.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{x}{5 x + 3} \cdot (5 x + 3)$

Step 5.3.11

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

Tap for more steps...

Step 5.3.11.1

Cancel the common factor.

$f (\frac{5 x - 6}{3 + 5 x}) = \frac{x}{5 x + 3} \cdot (5 x + 3)$

Step 5.3.11.2

Rewrite the expression.

$f (\frac{5 x - 6}{3 + 5 x}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x - 6}{3 + 5 x}$ is the inverse of $f (x) = \frac{- 3 x - 6}{5 x - 5}$ .

$f^{- 1} (x) = \frac{5 x - 6}{3 + 5 x}$