Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1}{y - 2} - 3 = x$ .

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

Step 2.2

Add $3$ to both sides of the equation.

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

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

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

$y - 2, 1, 1$

Step 2.3.2

Remove parentheses.

$y - 2, 1, 1$

Step 2.3.3

The LCM of one and any expression is the expression.

$y - 2$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $y - 2$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Rewrite the expression.

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Apply the distributive property.

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

Step 2.4.3.1.2

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

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

Step 2.4.3.1.3

Apply the distributive property.

$1 = x y - 2 x + 3 y + 3 \cdot - 2$

Step 2.4.3.1.4

Multiply $3$ by $- 2$ .

$1 = x y - 2 x + 3 y - 6$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $x y - 2 x + 3 y - 6 = 1$ .

$x y - 2 x + 3 y - 6 = 1$

Step 2.5.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.5.2.1

Add $2 x$ to both sides of the equation.

$x y + 3 y - 6 = 1 + 2 x$

Step 2.5.2.2

Add $6$ to both sides of the equation.

$x y + 3 y = 1 + 2 x + 6$

Step 2.5.2.3

Add $1$ and $6$ .

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

Step 2.5.3

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

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Step 2.5.3.1

Factor $y$ out of $x y$ .

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

Step 2.5.3.2

Factor $y$ out of $3 y$ .

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

Step 2.5.3.3

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

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

Step 2.5.4

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

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Step 2.5.4.1

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

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

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $x + 3$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

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

Step 2.5.4.2.1.2

Divide $y$ by $1$ .

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

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

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

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Step 4.2.3.1

Multiply $\frac{2 (\frac{1}{x - 2} - 3) + 7}{\frac{1}{x - 2} - 3 + 3}$ by $\frac{x - 2}{x - 2}$ .

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

Step 4.2.3.2

Combine.

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

Step 4.2.4

Apply the distributive property.

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

Step 4.2.5

Cancel the common factor of $x - 2$ .

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Step 4.2.5.1

Cancel the common factor.

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

Step 4.2.5.2

Rewrite the expression.

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

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Factor $x - 2$ out of $(x - 2) \cdot 2 (\frac{1}{x - 2} - 3) + (x - 2) \cdot 7$ .

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Step 4.2.6.1.1

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

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

Step 4.2.6.1.2

Factor $x - 2$ out of $(x - 2) \cdot 7$ .

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

Step 4.2.6.1.3

Factor $x - 2$ out of $(x - 2) (2 (\frac{1}{x - 2} - 3)) + (x - 2) (7)$ .

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

Step 4.2.6.2

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

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

Step 4.2.6.3

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

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

Step 4.2.6.4

Combine the numerators over the common denominator.

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

Step 4.2.6.5

Simplify the numerator.

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Step 4.2.6.5.1

Apply the distributive property.

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

Step 4.2.6.5.2

Multiply $- 3$ by $- 2$ .

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

Step 4.2.6.5.3

Add $1$ and $6$ .

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

Step 4.2.6.6

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

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

Step 4.2.6.7

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

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

Step 4.2.6.8

Combine the numerators over the common denominator.

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

Step 4.2.6.9

Reorder terms.

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

Step 4.2.6.10

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

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Step 4.2.6.10.1

Apply the distributive property.

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

Step 4.2.6.10.2

Multiply $7$ by $- 2$ .

$f^{- 1} (\frac{1}{x - 2} - 3) = \frac{(x - 2) (\frac{7 x - 14 + 2 (- 3 x + 7)}{x - 2})}{1 + (x - 2) \cdot - 3 + (x - 2) \cdot 3}$

Step 4.2.6.10.3

Apply the distributive property.

$f^{- 1} (\frac{1}{x - 2} - 3) = \frac{(x - 2) (\frac{7 x - 14 + 2 (- 3 x) + 2 \cdot 7}{x - 2})}{1 + (x - 2) \cdot - 3 + (x - 2) \cdot 3}$

Step 4.2.6.10.4

Multiply $- 3$ by $2$ .

$f^{- 1} (\frac{1}{x - 2} - 3) = \frac{(x - 2) (\frac{7 x - 14 - 6 x + 2 \cdot 7}{x - 2})}{1 + (x - 2) \cdot - 3 + (x - 2) \cdot 3}$

Step 4.2.6.10.5

Multiply $2$ by $7$ .

$f^{- 1} (\frac{1}{x - 2} - 3) = \frac{(x - 2) (\frac{7 x - 14 - 6 x + 14}{x - 2})}{1 + (x - 2) \cdot - 3 + (x - 2) \cdot 3}$

Step 4.2.6.10.6

Subtract $6 x$ from $7 x$ .

$f^{- 1} (\frac{1}{x - 2} - 3) = \frac{(x - 2) (\frac{x - 14 + 14}{x - 2})}{1 + (x - 2) \cdot - 3 + (x - 2) \cdot 3}$

Step 4.2.6.10.7

Add $- 14$ and $14$ .

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

Step 4.2.6.10.8

Add $x$ and $0$ .

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

Step 4.2.7

Simplify the denominator.

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Step 4.2.7.1

Apply the distributive property.

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

Step 4.2.7.2

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

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

Step 4.2.7.3

Multiply $- 2$ by $- 3$ .

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

Step 4.2.7.4

Apply the distributive property.

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

Step 4.2.7.5

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

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

Step 4.2.7.6

Multiply $- 2$ by $3$ .

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

Step 4.2.7.7

Add $1$ and $6$ .

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

Step 4.2.7.8

Add $- 3 x$ and $3 x$ .

$f^{- 1} (\frac{1}{x - 2} - 3) = \frac{(x - 2) (\frac{x}{x - 2})}{0 + 7 - 6}$

Step 4.2.7.9

Add $0$ and $7$ .

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

Step 4.2.7.10

Subtract $6$ from $7$ .

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

Step 4.2.8

Reduce the expression by cancelling the common factors.

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Step 4.2.8.1

Divide $(x - 2) \frac{x}{x - 2}$ by $1$ .

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

Step 4.2.8.2

Cancel the common factor of $x - 2$ .

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Step 4.2.8.2.1

Cancel the common factor.

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

Step 4.2.8.2.2

Rewrite the expression.

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

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

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Simplify the denominator.

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Step 4.3.3.1.1

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

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

Step 4.3.3.1.2

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

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

Step 4.3.3.1.3

Combine the numerators over the common denominator.

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

Step 4.3.3.1.4

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

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Step 4.3.3.1.4.1

Apply the distributive property.

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

Step 4.3.3.1.4.2

Multiply $- 2$ by $3$ .

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

Step 4.3.3.1.4.3

Subtract $2 x$ from $2 x$ .

$f (\frac{2 x + 7}{x + 3}) = \frac{1}{\frac{0 + 7 - 6}{x + 3}} - 3$

Step 4.3.3.1.4.4

Add $0$ and $7$ .

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

Step 4.3.3.1.4.5

Subtract $6$ from $7$ .

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

Step 4.3.3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.3.3

Multiply $x + 3$ by $1$ .

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

Step 4.3.4

Combine the opposite terms in $x + 3 - 3$ .

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Step 4.3.4.1

Subtract $3$ from $3$ .

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

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

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

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