Trigonometry Examples

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

Step 1

Write $f (x) = \frac{4}{x - 2} + 1$ as an equation.

$y = \frac{4}{x - 2} + 1$

Step 2

Interchange the variables.

$x = \frac{4}{y - 2} + 1$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{4}{y - 2} + 1 = x$ .

$\frac{4}{y - 2} + 1 = x$

Step 3.2

Subtract $1$ from both sides of the equation.

$\frac{4}{y - 2} = x - 1$

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 $\frac{4}{y - 2} = x - 1$ by $y - 2$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{4}{y - 2} = x - 1$ by $y - 2$ .

$\frac{4}{y - 2} (y - 2) = x (y - 2) - (y - 2)$

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.

$\frac{4}{y - 2} (y - 2) = x (y - 2) - (y - 2)$

Step 3.4.2.1.2

Rewrite the expression.

$4 = x (y - 2) - (y - 2)$

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.

$4 = x y + x \cdot - 2 - (y - 2)$

Step 3.4.3.1.2

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

$4 = x y - 2 \cdot x - (y - 2)$

Step 3.4.3.1.3

Apply the distributive property.

$4 = x y - 2 x - y - - 2$

Step 3.4.3.1.4

Multiply $- 1$ by $- 2$ .

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

Step 3.5

Solve the equation.

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Step 3.5.1

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

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

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.

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

Step 3.5.2.2

Subtract $2$ from both sides of the equation.

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

Step 3.5.2.3

Subtract $2$ from $4$ .

$x y - y = 2 x + 2$

Step 3.5.3

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

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Step 3.5.3.1

Factor $y$ out of $x y$ .

$y x - y = 2 x + 2$

Step 3.5.3.2

Factor $y$ out of $- y$ .

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

Step 3.5.3.3

Factor $y$ out of $y x + y \cdot - 1$ .

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

Step 3.5.4

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

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Step 3.5.4.1

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

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

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $x - 1$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

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

Step 3.5.4.2.1.2

Divide $y$ by $1$ .

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

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.

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

Step 3.5.4.3.2

Factor $2$ out of $2 x + 2$ .

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Step 3.5.4.3.2.1

Factor $2$ out of $2 x$ .

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

Step 3.5.4.3.2.2

Factor $2$ out of $2$ .

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

Step 3.5.4.3.2.3

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

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{2 (x + 1)}{x - 1}$ is the inverse of $f (x) = \frac{4}{x - 2} + 1$ .

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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 $f^{- 1} (\frac{4}{x - 2} + 1)$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

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

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

Step 5.2.3.2

Combine the numerators over the common denominator.

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

Step 5.2.3.3

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

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

Step 5.2.3.4

Combine the numerators over the common denominator.

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

Step 5.2.3.5

Reorder terms.

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

Step 5.2.3.6

Rewrite $\frac{x + x + 4 - 2 - 2}{x - 2}$ in a factored form.

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Step 5.2.3.6.1

Add $x$ and $x$ .

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

Step 5.2.3.6.2

Subtract $2$ from $4$ .

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

Step 5.2.3.6.3

Subtract $2$ from $2$ .

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

Step 5.2.3.6.4

Add $2 x$ and $0$ .

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

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

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

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

Step 5.2.4.2

Combine the numerators over the common denominator.

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

Step 5.2.4.3

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

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

Step 5.2.4.4

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

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

Step 5.2.4.5

Combine the numerators over the common denominator.

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

Step 5.2.4.6

Reorder terms.

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

Step 5.2.4.7

Rewrite $\frac{x + 4 - 2 - (x - 2)}{x - 2}$ in a factored form.

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Step 5.2.4.7.1

Apply the distributive property.

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

Step 5.2.4.7.2

Multiply $- 1$ by $- 2$ .

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

Step 5.2.4.7.3

Subtract $x$ from $x$ .

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

Step 5.2.4.7.4

Add $0$ and $4$ .

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

Step 5.2.4.7.5

Subtract $2$ from $4$ .

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

Step 5.2.4.7.6

Add $2$ and $2$ .

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

Step 5.2.5

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

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

Step 5.2.6

Multiply $2$ by $2$ .

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

Step 5.2.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.8

Cancel the common factor of $4$ .

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Step 5.2.8.1

Factor $4$ out of $4 x$ .

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

Step 5.2.8.2

Cancel the common factor.

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

Step 5.2.8.3

Rewrite the expression.

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

Step 5.2.9

Cancel the common factor of $x - 2$ .

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Step 5.2.9.1

Cancel the common factor.

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

Step 5.2.9.2

Rewrite the expression.

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Cancel the common factor of $4$ and $(\frac{2 (x + 1)}{x - 1}) - 2$ .

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Step 5.3.3.1.1

Factor $2$ out of $4$ .

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

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

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

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

Step 5.3.3.1.2.2

Factor $2$ out of $- 2$ .

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

Step 5.3.3.1.2.3

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

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

Step 5.3.3.1.2.4

Cancel the common factor.

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

Step 5.3.3.1.2.5

Rewrite the expression.

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

Step 5.3.3.2

Simplify the denominator.

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Step 5.3.3.2.1

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

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

Step 5.3.3.2.2

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

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

Step 5.3.3.2.3

Combine the numerators over the common denominator.

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

Step 5.3.3.2.4

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

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Step 5.3.3.2.4.1

Apply the distributive property.

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

Step 5.3.3.2.4.2

Multiply $- 1$ by $- 1$ .

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

Step 5.3.3.2.4.3

Subtract $x$ from $x$ .

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

Step 5.3.3.2.4.4

Add $0$ and $1$ .

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

Step 5.3.3.2.4.5

Add $1$ and $1$ .

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

Step 5.3.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.3.4

Cancel the common factor of $2$ .

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Step 5.3.3.4.1

Cancel the common factor.

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

Step 5.3.3.4.2

Rewrite the expression.

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

Step 5.3.4

Combine the opposite terms in $x - 1 + 1$ .

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Step 5.3.4.1

Add $- 1$ and $1$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 (x + 1)}{x - 1}$ is the inverse of $f (x) = \frac{4}{x - 2} + 1$ .

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