Trigonometry Examples

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

Step 1

Write $f (x) = \frac{3}{x - 2} + 2$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Subtract $2$ from both sides of the equation.

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

Step 3.3

Find the LCD of the terms in the equation.

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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.

$y - 2, 1, 1$

Step 3.3.2

Remove parentheses.

$y - 2, 1, 1$

Step 3.3.3

The LCM of one and any expression is the expression.

$y - 2$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $y - 2$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Rewrite the expression.

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Apply the distributive property.

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

Step 3.4.3.1.2

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

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

Step 3.4.3.1.3

Apply the distributive property.

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

Step 3.4.3.1.4

Multiply $- 2$ by $- 2$ .

$3 = x y - 2 x - 2 y + 4$

Step 3.5

Solve the equation.

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Step 3.5.1

Rewrite the equation as $x y - 2 x - 2 y + 4 = 3$ .

$x y - 2 x - 2 y + 4 = 3$

Step 3.5.2

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

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Step 3.5.2.1

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

$x y - 2 y + 4 = 3 + 2 x$

Step 3.5.2.2

Subtract $4$ from both sides of the equation.

$x y - 2 y = 3 + 2 x - 4$

Step 3.5.2.3

Subtract $4$ from $3$ .

$x y - 2 y = 2 x - 1$

Step 3.5.3

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

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Step 3.5.3.1

Factor $y$ out of $x y$ .

$y x - 2 y = 2 x - 1$

Step 3.5.3.2

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

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

Step 3.5.3.3

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

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

Step 3.5.4

Divide each term in $y (x - 2) = 2 x - 1$ by $x - 2$ and simplify.

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Step 3.5.4.1

Divide each term in $y (x - 2) = 2 x - 1$ by $x - 2$ .

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

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $x - 2$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

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

Step 3.5.4.2.1.2

Divide $y$ by $1$ .

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

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

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

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Step 5.2.3.1

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

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

Step 5.2.3.2

Combine.

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

Step 5.2.4

Apply the distributive property.

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

Step 5.2.5

Cancel the common factor of $x - 2$ .

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Step 5.2.5.1

Cancel the common factor.

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

Step 5.2.5.2

Rewrite the expression.

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

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

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

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Step 5.2.6.1.1

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

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

Step 5.2.6.1.2

Factor $x - 2$ out of $(x - 2) \cdot - 1$ .

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

Step 5.2.6.1.3

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

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

Step 5.2.6.2

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

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

Step 5.2.6.3

Combine the numerators over the common denominator.

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

Step 5.2.6.4

Simplify the numerator.

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Step 5.2.6.4.1

Apply the distributive property.

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

Step 5.2.6.4.2

Multiply $2$ by $- 2$ .

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (2 (\frac{3 + 2 x - 4}{x - 2}) - 1)}{3 + (x - 2) \cdot 2 + (x - 2) \cdot - 2}$

Step 5.2.6.4.3

Subtract $4$ from $3$ .

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

Step 5.2.6.5

Combine $2$ and $\frac{2 x - 1}{x - 2}$ .

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

Step 5.2.6.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 5.2.6.7

Combine $- 1$ and $\frac{x - 2}{x - 2}$ .

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

Step 5.2.6.8

Combine the numerators over the common denominator.

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

Step 5.2.6.9

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

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Step 5.2.6.9.1

Apply the distributive property.

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

Step 5.2.6.9.2

Multiply $2$ by $2$ .

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{4 x + 2 \cdot - 1 - (x - 2)}{x - 2})}{3 + (x - 2) \cdot 2 + (x - 2) \cdot - 2}$

Step 5.2.6.9.3

Multiply $2$ by $- 1$ .

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{4 x - 2 - (x - 2)}{x - 2})}{3 + (x - 2) \cdot 2 + (x - 2) \cdot - 2}$

Step 5.2.6.9.4

Apply the distributive property.

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{4 x - 2 - x + 2}{x - 2})}{3 + (x - 2) \cdot 2 + (x - 2) \cdot - 2}$

Step 5.2.6.9.5

Multiply $- 1$ by $- 2$ .

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{4 x - 2 - x + 2}{x - 2})}{3 + (x - 2) \cdot 2 + (x - 2) \cdot - 2}$

Step 5.2.6.9.6

Subtract $x$ from $4 x$ .

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

Step 5.2.6.9.7

Add $- 2$ and $2$ .

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

Step 5.2.6.9.8

Add $3 x$ and $0$ .

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

Step 5.2.7

Simplify the denominator.

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Step 5.2.7.1

Apply the distributive property.

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

Step 5.2.7.2

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

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

Step 5.2.7.3

Multiply $- 2$ by $2$ .

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{3 x}{x - 2})}{3 + 2 \cdot x - 4 + (x - 2) \cdot - 2}$

Step 5.2.7.4

Apply the distributive property.

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{3 x}{x - 2})}{3 + 2 x - 4 + x \cdot - 2 - 2 \cdot - 2}$

Step 5.2.7.5

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

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{3 x}{x - 2})}{3 + 2 x - 4 - 2 \cdot x - 2 \cdot - 2}$

Step 5.2.7.6

Multiply $- 2$ by $- 2$ .

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{3 x}{x - 2})}{3 + 2 x - 4 - 2 \cdot x + 4}$

Step 5.2.7.7

Subtract $4$ from $3$ .

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{3 x}{x - 2})}{2 x - 1 - 2 x + 4}$

Step 5.2.7.8

Subtract $2 x$ from $2 x$ .

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{3 x}{x - 2})}{0 - 1 + 4}$

Step 5.2.7.9

Subtract $1$ from $0$ .

$f^{- 1} (\frac{3}{x - 2} + 2) = \frac{(x - 2) (\frac{3 x}{x - 2})}{- 1 + 4}$

Step 5.2.7.10

Add $- 1$ and $4$ .

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

Step 5.2.8

Reduce the expression by cancelling the common factors.

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Step 5.2.8.1

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

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

Step 5.2.8.2

Cancel the common factor of $3$ .

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Step 5.2.8.2.1

Factor $3$ out of $3 x$ .

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

Step 5.2.8.2.2

Cancel the common factor.

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

Step 5.2.8.2.3

Rewrite the expression.

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

Step 5.2.8.3

Cancel the common factor of $x - 2$ .

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Step 5.2.8.3.1

Cancel the common factor.

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

Step 5.2.8.3.2

Rewrite the expression.

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the denominator.

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Step 5.3.3.1.1

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

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

Step 5.3.3.1.2

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

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

Step 5.3.3.1.3

Combine the numerators over the common denominator.

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

Step 5.3.3.1.4

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

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Step 5.3.3.1.4.1

Apply the distributive property.

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

Step 5.3.3.1.4.2

Multiply $- 2$ by $- 2$ .

$f (\frac{2 x - 1}{x - 2}) = \frac{3}{\frac{2 x - 1 - 2 x + 4}{x - 2}} + 2$

Step 5.3.3.1.4.3

Subtract $2 x$ from $2 x$ .

$f (\frac{2 x - 1}{x - 2}) = \frac{3}{\frac{0 - 1 + 4}{x - 2}} + 2$

Step 5.3.3.1.4.4

Subtract $1$ from $0$ .

$f (\frac{2 x - 1}{x - 2}) = \frac{3}{\frac{- 1 + 4}{x - 2}} + 2$

Step 5.3.3.1.4.5

Add $- 1$ and $4$ .

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

Step 5.3.3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.3.1

Cancel the common factor.

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

Step 5.3.3.3.2

Rewrite the expression.

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

Step 5.3.4

Combine the opposite terms in $x - 2 + 2$ .

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Step 5.3.4.1

Add $- 2$ and $2$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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