Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

Subtract $1$ from both sides of the equation.

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

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.

$y - 1, 1, 1$

Step 2.3.2

Remove parentheses.

$y - 1, 1, 1$

Step 2.3.3

The LCM of one and any expression is the expression.

$y - 1$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $y - 1$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Rewrite the expression.

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

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.

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

Step 2.4.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 2.4.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.4.3.1.4

Apply the distributive property.

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

Step 2.4.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.5

Solve the equation.

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Step 2.5.1

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

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

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 $x$ to both sides of the equation.

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

Step 2.5.2.2

Subtract $1$ from both sides of the equation.

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

Step 2.5.2.3

Subtract $1$ from $2$ .

$x y - y = x + 1$

Step 2.5.3

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

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Step 2.5.3.1

Factor $y$ out of $x y$ .

$y x - y = x + 1$

Step 2.5.3.2

Factor $y$ out of $- y$ .

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

Step 2.5.3.3

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

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

Step 2.5.4

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

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Step 2.5.4.1

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

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

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $x - 1$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

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

Step 2.5.4.2.1.2

Divide $y$ by $1$ .

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

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.

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

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Multiply the numerator and denominator of the fraction by $x - 1$ .

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Step 4.2.3.1

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

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

Step 4.2.3.2

Combine.

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

Step 4.2.4

Apply the distributive property.

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

Step 4.2.5

Simplify by cancelling.

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Step 4.2.5.1

Cancel the common factor of $x - 1$ .

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Step 4.2.5.1.1

Cancel the common factor.

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

Step 4.2.5.1.2

Rewrite the expression.

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

Step 4.2.5.2

Cancel the common factor of $x - 1$ .

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Step 4.2.5.2.1

Cancel the common factor.

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

Step 4.2.5.2.2

Rewrite the expression.

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

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $x - 1$ by $1$ .

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

Step 4.2.6.2

Multiply $x - 1$ by $1$ .

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

Step 4.2.6.3

Subtract $1$ from $2$ .

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

Step 4.2.6.4

Add $x$ and $x$ .

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

Step 4.2.6.5

Subtract $1$ from $1$ .

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

Step 4.2.6.6

Add $2 x$ and $0$ .

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

Step 4.2.7

Simplify the denominator.

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Step 4.2.7.1

Multiply $x - 1$ by $1$ .

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

Step 4.2.7.2

Apply the distributive property.

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

Step 4.2.7.3

Move $- 1$ to the left of $x$ .

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

Step 4.2.7.4

Multiply $- 1$ by $- 1$ .

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

Step 4.2.7.5

Rewrite $- 1 x$ as $- x$ .

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

Step 4.2.7.6

Subtract $1$ from $2$ .

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

Step 4.2.7.7

Subtract $x$ from $x$ .

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

Step 4.2.7.8

Add $0$ and $1$ .

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

Step 4.2.7.9

Add $1$ and $1$ .

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

Step 4.2.8

Cancel the common factor of $2$ .

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Step 4.2.8.1

Cancel the common factor.

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

Step 4.2.8.2

Divide $x$ by $1$ .

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

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

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

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

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

Step 4.3.3.1.2

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

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

Step 4.3.3.1.3

Combine the numerators over the common denominator.

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

Step 4.3.3.1.4

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

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Step 4.3.3.1.4.1

Apply the distributive property.

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

Step 4.3.3.1.4.2

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3.1.4.3

Subtract $x$ from $x$ .

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

Step 4.3.3.1.4.4

Add $0$ and $1$ .

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

Step 4.3.3.1.4.5

Add $1$ and $1$ .

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

Step 4.3.3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.3.1

Cancel the common factor.

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

Step 4.3.3.3.2

Rewrite the expression.

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

Step 4.3.4

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

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Step 4.3.4.1

Add $- 1$ and $1$ .

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

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

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

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