Trigonometry Examples

$\frac{3 x - 5}{2 x + 7}$

Step 1

Interchange the variables.

$x = \frac{3 y - 5}{2 y + 7}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{3 y - 5}{2 y + 7} = x$ .

$\frac{3 y - 5}{2 y + 7} = x$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$2 y + 7, 1$

Step 2.2.2

Remove parentheses.

$2 y + 7, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$2 y + 7$

Step 2.3

Multiply each term in $\frac{3 y - 5}{2 y + 7} = x$ by $2 y + 7$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{3 y - 5}{2 y + 7} = x$ by $2 y + 7$ .

$\frac{3 y - 5}{2 y + 7} (2 y + 7) = x (2 y + 7)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2 y + 7$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 y - 5}{2 y + 7} (2 y + 7) = x (2 y + 7)$

Step 2.3.2.1.2

Rewrite the expression.

$3 y - 5 = x (2 y + 7)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$3 y - 5 = x (2 y) + x \cdot 7$

Step 2.3.3.2

Reorder.

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Step 2.3.3.2.1

Rewrite using the commutative property of multiplication.

$3 y - 5 = 2 x y + x \cdot 7$

Step 2.3.3.2.2

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

$3 y - 5 = 2 x y + 7 x$

Step 2.4

Solve the equation.

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Step 2.4.1

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

$3 y - 5 - 2 x y = 7 x$

Step 2.4.2

Add $5$ to both sides of the equation.

$3 y - 2 x y = 7 x + 5$

Step 2.4.3

Factor $y$ out of $3 y - 2 x y$ .

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Step 2.4.3.1

Factor $y$ out of $3 y$ .

$y \cdot 3 - 2 x y = 7 x + 5$

Step 2.4.3.2

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

$y \cdot 3 + y (- 2 x) = 7 x + 5$

Step 2.4.3.3

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

$y (3 - 2 x) = 7 x + 5$

Step 2.4.4

Divide each term in $y (3 - 2 x) = 7 x + 5$ by $3 - 2 x$ and simplify.

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Step 2.4.4.1

Divide each term in $y (3 - 2 x) = 7 x + 5$ by $3 - 2 x$ .

$\frac{y (3 - 2 x)}{3 - 2 x} = \frac{7 x}{3 - 2 x} + \frac{5}{3 - 2 x}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $3 - 2 x$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{y (3 - 2 x)}{3 - 2 x} = \frac{7 x}{3 - 2 x} + \frac{5}{3 - 2 x}$

Step 2.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{7 x}{3 - 2 x} + \frac{5}{3 - 2 x}$

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.

$y = \frac{7 x + 5}{3 - 2 x}$

Step 3

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

$f^{- 1} (x) = \frac{7 x + 5}{3 - 2 x}$

Step 4

Verify if $f^{- 1} (x) = \frac{7 x + 5}{3 - 2 x}$ is the inverse of $f (x) = \frac{3 x - 5}{2 x + 7}$ .

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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 $f^{- 1} (\frac{3 x - 5}{2 x + 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{7 (\frac{3 x - 5}{2 x + 7}) + 5}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Combine $7$ and $\frac{3 x - 5}{2 x + 7}$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{7 (3 x - 5)}{2 x + 7} + 5}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2 x + 7}{2 x + 7}$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{7 (3 x - 5)}{2 x + 7} + \frac{5 (2 x + 7)}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{7 (3 x - 5) + 5 (2 x + 7)}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.4

Rewrite $\frac{7 (3 x - 5) + 5 (2 x + 7)}{2 x + 7}$ in a factored form.

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Step 4.2.3.4.1

Apply the distributive property.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{7 (3 x) + 7 \cdot - 5 + 5 (2 x + 7)}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.4.2

Multiply $3$ by $7$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{21 x + 7 \cdot - 5 + 5 (2 x + 7)}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.4.3

Multiply $7$ by $- 5$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{21 x - 35 + 5 (2 x + 7)}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.4.4

Apply the distributive property.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{21 x - 35 + 5 (2 x) + 5 \cdot 7}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.4.5

Multiply $2$ by $5$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{21 x - 35 + 10 x + 5 \cdot 7}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.4.6

Multiply $5$ by $7$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{21 x - 35 + 10 x + 35}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.4.7

Add $21 x$ and $10 x$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x - 35 + 35}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.4.8

Add $- 35$ and $35$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x + 0}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.3.4.9

Add $31 x$ and $0$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{3 - 2 \frac{3 x - 5}{2 x + 7}}$

Step 4.2.4

Simplify the denominator.

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Step 4.2.4.1

Combine $- 2$ and $\frac{3 x - 5}{2 x + 7}$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{3 + \frac{- 2 (3 x - 5)}{2 x + 7}}$

Step 4.2.4.2

Move the negative in front of the fraction.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{3 - \frac{2 (3 x - 5)}{2 x + 7}}$

Step 4.2.4.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2 x + 7}{2 x + 7}$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{3 (2 x + 7)}{2 x + 7} - \frac{2 (3 x - 5)}{2 x + 7}}$

Step 4.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{3 (2 x + 7) - 2 (3 x - 5)}{2 x + 7}}$

Step 4.2.4.5

Rewrite $\frac{3 (2 x + 7) - 2 (3 x - 5)}{2 x + 7}$ in a factored form.

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Step 4.2.4.5.1

Apply the distributive property.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{3 (2 x) + 3 \cdot 7 - 2 (3 x - 5)}{2 x + 7}}$

Step 4.2.4.5.2

Multiply $2$ by $3$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{6 x + 3 \cdot 7 - 2 (3 x - 5)}{2 x + 7}}$

Step 4.2.4.5.3

Multiply $3$ by $7$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{6 x + 21 - 2 (3 x - 5)}{2 x + 7}}$

Step 4.2.4.5.4

Apply the distributive property.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{6 x + 21 - 2 (3 x) - 2 \cdot - 5}{2 x + 7}}$

Step 4.2.4.5.5

Multiply $3$ by $- 2$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{6 x + 21 - 6 x - 2 \cdot - 5}{2 x + 7}}$

Step 4.2.4.5.6

Multiply $- 2$ by $- 5$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{6 x + 21 - 6 x + 10}{2 x + 7}}$

Step 4.2.4.5.7

Subtract $6 x$ from $6 x$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{0 + 21 + 10}{2 x + 7}}$

Step 4.2.4.5.8

Add $0$ and $21$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{21 + 10}{2 x + 7}}$

Step 4.2.4.5.9

Add $21$ and $10$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{\frac{31 x}{2 x + 7}}{\frac{31}{2 x + 7}}$

Step 4.2.5

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{31 x}{2 x + 7} \cdot \frac{2 x + 7}{31}$

Step 4.2.6

Cancel the common factor of $31$ .

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Step 4.2.6.1

Factor $31$ out of $31 x$ .

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{31 (x)}{2 x + 7} \cdot \frac{2 x + 7}{31}$

Step 4.2.6.2

Cancel the common factor.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{31 x}{2 x + 7} \cdot \frac{2 x + 7}{31}$

Step 4.2.6.3

Rewrite the expression.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{x}{2 x + 7} \cdot (2 x + 7)$

Step 4.2.7

Cancel the common factor of $2 x + 7$ .

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Step 4.2.7.1

Cancel the common factor.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = \frac{x}{2 x + 7} \cdot (2 x + 7)$

Step 4.2.7.2

Rewrite the expression.

$f^{- 1} (\frac{3 x - 5}{2 x + 7}) = x$

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 $f (\frac{7 x + 5}{3 - 2 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{3 (\frac{7 x + 5}{3 - 2 x}) - 5}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

Combine $3$ and $\frac{7 x + 5}{3 - 2 x}$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{3 (7 x + 5)}{3 - 2 x} - 5}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.2

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

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{3 (7 x + 5)}{3 - 2 x} - 5 \cdot \frac{3 - 2 x}{3 - 2 x}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.3

Combine $- 5$ and $\frac{3 - 2 x}{3 - 2 x}$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{3 (7 x + 5)}{3 - 2 x} + \frac{- 5 (3 - 2 x)}{3 - 2 x}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.4

Combine the numerators over the common denominator.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{3 (7 x + 5) - 5 (3 - 2 x)}{3 - 2 x}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.5

Reorder terms.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{3 (7 x + 5) - 5 (3 - 2 x)}{- 2 x + 3}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.6

Rewrite $\frac{3 (7 x + 5) - 5 (3 - 2 x)}{- 2 x + 3}$ in a factored form.

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Step 4.3.3.6.1

Apply the distributive property.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{3 (7 x) + 3 \cdot 5 - 5 (3 - 2 x)}{- 2 x + 3}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.6.2

Multiply $7$ by $3$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{21 x + 3 \cdot 5 - 5 (3 - 2 x)}{- 2 x + 3}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.6.3

Multiply $3$ by $5$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{21 x + 15 - 5 (3 - 2 x)}{- 2 x + 3}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.6.4

Apply the distributive property.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{21 x + 15 - 5 \cdot 3 - 5 (- 2 x)}{- 2 x + 3}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.6.5

Multiply $- 5$ by $3$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{21 x + 15 - 15 - 5 (- 2 x)}{- 2 x + 3}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.6.6

Multiply $- 2$ by $- 5$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{21 x + 15 - 15 + 10 x}{- 2 x + 3}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.6.7

Add $21 x$ and $10 x$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x + 15 - 15}{- 2 x + 3}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.6.8

Subtract $15$ from $15$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x + 0}{- 2 x + 3}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.3.6.9

Add $31 x$ and $0$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{2 (\frac{7 x + 5}{3 - 2 x}) + 7}$

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Combine $2$ and $\frac{7 x + 5}{3 - 2 x}$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{2 (7 x + 5)}{3 - 2 x} + 7}$

Step 4.3.4.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{3 - 2 x}{3 - 2 x}$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{2 (7 x + 5)}{3 - 2 x} + \frac{7 (3 - 2 x)}{3 - 2 x}}$

Step 4.3.4.3

Combine the numerators over the common denominator.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{2 (7 x + 5) + 7 (3 - 2 x)}{3 - 2 x}}$

Step 4.3.4.4

Rewrite $\frac{2 (7 x + 5) + 7 (3 - 2 x)}{3 - 2 x}$ in a factored form.

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Step 4.3.4.4.1

Apply the distributive property.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{2 (7 x) + 2 \cdot 5 + 7 (3 - 2 x)}{3 - 2 x}}$

Step 4.3.4.4.2

Multiply $7$ by $2$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{14 x + 2 \cdot 5 + 7 (3 - 2 x)}{3 - 2 x}}$

Step 4.3.4.4.3

Multiply $2$ by $5$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{14 x + 10 + 7 (3 - 2 x)}{3 - 2 x}}$

Step 4.3.4.4.4

Apply the distributive property.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{14 x + 10 + 7 \cdot 3 + 7 (- 2 x)}{3 - 2 x}}$

Step 4.3.4.4.5

Multiply $7$ by $3$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{14 x + 10 + 21 + 7 (- 2 x)}{3 - 2 x}}$

Step 4.3.4.4.6

Multiply $- 2$ by $7$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{14 x + 10 + 21 - 14 x}{3 - 2 x}}$

Step 4.3.4.4.7

Subtract $14 x$ from $14 x$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{0 + 10 + 21}{3 - 2 x}}$

Step 4.3.4.4.8

Add $0$ and $10$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{10 + 21}{3 - 2 x}}$

Step 4.3.4.4.9

Add $10$ and $21$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{\frac{31 x}{- 2 x + 3}}{\frac{31}{3 - 2 x}}$

Step 4.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{31 x}{- 2 x + 3} \cdot \frac{3 - 2 x}{31}$

Step 4.3.6

Cancel the common factor of $31$ .

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Step 4.3.6.1

Factor $31$ out of $31 x$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{31 (x)}{- 2 x + 3} \cdot \frac{3 - 2 x}{31}$

Step 4.3.6.2

Cancel the common factor.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{31 x}{- 2 x + 3} \cdot \frac{3 - 2 x}{31}$

Step 4.3.6.3

Rewrite the expression.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{x}{- 2 x + 3} \cdot (3 - 2 x)$

Step 4.3.7

Multiply $\frac{x}{- 2 x + 3}$ by $3 - 2 x$ .

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{x (3 - 2 x)}{- 2 x + 3}$

Step 4.3.8

Cancel the common factor of $3 - 2 x$ and $- 2 x + 3$ .

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Step 4.3.8.1

Reorder terms.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{x (- 2 x + 3)}{- 2 x + 3}$

Step 4.3.8.2

Cancel the common factor.

$f (\frac{7 x + 5}{3 - 2 x}) = \frac{x (- 2 x + 3)}{- 2 x + 3}$

Step 4.3.8.3

Divide $x$ by $1$ .

$f (\frac{7 x + 5}{3 - 2 x}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x + 5}{3 - 2 x}$ is the inverse of $f (x) = \frac{3 x - 5}{2 x + 7}$ .

$f^{- 1} (x) = \frac{7 x + 5}{3 - 2 x}$