Trigonometry Examples

$\frac{x}{8 - 9 x}$

Step 1

Interchange the variables.

$x = \frac{y}{8 - 9 y}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{y}{8 - 9 y} = x$ .

$\frac{y}{8 - 9 y} = x$

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.

$8 - 9 y, 1$

Step 2.2.2

Remove parentheses.

$8 - 9 y, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$8 - 9 y$

Step 2.3

Multiply each term in $\frac{y}{8 - 9 y} = x$ by $8 - 9 y$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{y}{8 - 9 y} = x$ by $8 - 9 y$ .

$\frac{y}{8 - 9 y} (8 - 9 y) = x (8 - 9 y)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $8 - 9 y$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{y}{8 - 9 y} (8 - 9 y) = x (8 - 9 y)$

Step 2.3.2.1.2

Rewrite the expression.

$y = x (8 - 9 y)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$y = x \cdot 8 + x (- 9 y)$

Step 2.3.3.2

Reorder.

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Step 2.3.3.2.1

Move $8$ to the left of $x$ .

$y = 8 \cdot x + x (- 9 y)$

Step 2.3.3.2.2

Rewrite using the commutative property of multiplication.

$y = 8 x - 9 x y$

Step 2.4

Solve the equation.

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Step 2.4.1

Add $9 x y$ to both sides of the equation.

$y + 9 x y = 8 x$

Step 2.4.2

Factor $y$ out of $y + 9 x y$ .

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Step 2.4.2.1

Factor $y$ out of $y^{1}$ .

$y \cdot 1 + 9 x y = 8 x$

Step 2.4.2.2

Factor $y$ out of $9 x y$ .

$y \cdot 1 + y (9 x) = 8 x$

Step 2.4.2.3

Factor $y$ out of $y \cdot 1 + y (9 x)$ .

$y (1 + 9 x) = 8 x$

Step 2.4.3

Divide each term in $y (1 + 9 x) = 8 x$ by $1 + 9 x$ and simplify.

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Step 2.4.3.1

Divide each term in $y (1 + 9 x) = 8 x$ by $1 + 9 x$ .

$\frac{y (1 + 9 x)}{1 + 9 x} = \frac{8 x}{1 + 9 x}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $1 + 9 x$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{y (1 + 9 x)}{1 + 9 x} = \frac{8 x}{1 + 9 x}$

Step 2.4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{8 x}{1 + 9 x}$

Step 3

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

$f^{- 1} (x) = \frac{8 x}{1 + 9 x}$

Step 4

Verify if $f^{- 1} (x) = \frac{8 x}{1 + 9 x}$ is the inverse of $f (x) = \frac{x}{8 - 9 x}$ .

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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 $f^{- 1} (\frac{x}{8 - 9 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{8 (\frac{x}{8 - 9 x})}{1 + 9 (\frac{x}{8 - 9 x})}$

Step 4.2.3

Combine $8$ and $\frac{x}{8 - 9 x}$ .

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{\frac{8 x}{8 - 9 x}}{1 + 9 (\frac{x}{8 - 9 x})}$

Step 4.2.4

Simplify the denominator.

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Step 4.2.4.1

Combine $9$ and $\frac{x}{8 - 9 x}$ .

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{\frac{8 x}{8 - 9 x}}{1 + \frac{9 x}{8 - 9 x}}$

Step 4.2.4.2

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

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{\frac{8 x}{8 - 9 x}}{\frac{8 - 9 x}{8 - 9 x} + \frac{9 x}{8 - 9 x}}$

Step 4.2.4.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{\frac{8 x}{8 - 9 x}}{\frac{8 - 9 x + 9 x}{8 - 9 x}}$

Step 4.2.4.4

Rewrite $\frac{8 - 9 x + 9 x}{8 - 9 x}$ in a factored form.

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Step 4.2.4.4.1

Add $- 9 x$ and $9 x$ .

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{\frac{8 x}{8 - 9 x}}{\frac{8 + 0}{8 - 9 x}}$

Step 4.2.4.4.2

Add $8$ and $0$ .

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{\frac{8 x}{8 - 9 x}}{\frac{8}{8 - 9 x}}$

Step 4.2.5

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{8 x}{8 - 9 x} \cdot \frac{8 - 9 x}{8}$

Step 4.2.6

Cancel the common factor of $8$ .

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Step 4.2.6.1

Factor $8$ out of $8 x$ .

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{8 (x)}{8 - 9 x} \cdot \frac{8 - 9 x}{8}$

Step 4.2.6.2

Cancel the common factor.

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{8 x}{8 - 9 x} \cdot \frac{8 - 9 x}{8}$

Step 4.2.6.3

Rewrite the expression.

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{x}{8 - 9 x} \cdot (8 - 9 x)$

Step 4.2.7

Cancel the common factor of $8 - 9 x$ .

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Step 4.2.7.1

Cancel the common factor.

$f^{- 1} (\frac{x}{8 - 9 x}) = \frac{x}{8 - 9 x} \cdot (8 - 9 x)$

Step 4.2.7.2

Rewrite the expression.

$f^{- 1} (\frac{x}{8 - 9 x}) = x$

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 $f (\frac{8 x}{1 + 9 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{\frac{8 x}{1 + 9 x}}{8 - 9 \frac{8 x}{1 + 9 x}}$

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{8 - 9 \frac{8 x}{1 + 9 x}}$

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Multiply $- 9 \frac{8 x}{1 + 9 x}$ .

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Step 4.3.4.1.1

Combine $- 9$ and $\frac{8 x}{1 + 9 x}$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{8 + \frac{- 9 (8 x)}{1 + 9 x}}$

Step 4.3.4.1.2

Multiply $8$ by $- 9$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{8 + \frac{- 72 x}{1 + 9 x}}$

Step 4.3.4.2

Move the negative in front of the fraction.

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{8 - \frac{72 x}{1 + 9 x}}$

Step 4.3.4.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{1 + 9 x}{1 + 9 x}$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{\frac{8 (1 + 9 x)}{1 + 9 x} - \frac{72 x}{1 + 9 x}}$

Step 4.3.4.4

Combine the numerators over the common denominator.

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{\frac{8 (1 + 9 x) - 72 x}{1 + 9 x}}$

Step 4.3.4.5

Rewrite $\frac{8 (1 + 9 x) - 72 x}{1 + 9 x}$ in a factored form.

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Step 4.3.4.5.1

Factor $8$ out of $8 (1 + 9 x) - 72 x$ .

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Step 4.3.4.5.1.1

Factor $8$ out of $- 72 x$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{\frac{8 (1 + 9 x) + 8 (- 9 x)}{1 + 9 x}}$

Step 4.3.4.5.1.2

Factor $8$ out of $8 (1 + 9 x) + 8 (- 9 x)$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{\frac{8 (1 + 9 x - 9 x)}{1 + 9 x}}$

Step 4.3.4.5.2

Subtract $9 x$ from $9 x$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{\frac{8 (1 + 0)}{1 + 9 x}}$

Step 4.3.4.5.3

Add $1$ and $0$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{\frac{8 \cdot 1}{1 + 9 x}}$

Step 4.3.4.6

Multiply $8$ by $1$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1}{\frac{8}{1 + 9 x}}$

Step 4.3.5

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot (1 (\frac{1 + 9 x}{8}))$

Step 4.3.6

Multiply $\frac{1 + 9 x}{8}$ by $1$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1 + 9 x}{8}$

Step 4.3.7

Cancel the common factor of $8$ .

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Step 4.3.7.1

Factor $8$ out of $8 x$ .

$f (\frac{8 x}{1 + 9 x}) = \frac{8 (x)}{1 + 9 x} \cdot \frac{1 + 9 x}{8}$

Step 4.3.7.2

Cancel the common factor.

$f (\frac{8 x}{1 + 9 x}) = \frac{8 x}{1 + 9 x} \cdot \frac{1 + 9 x}{8}$

Step 4.3.7.3

Rewrite the expression.

$f (\frac{8 x}{1 + 9 x}) = \frac{x}{1 + 9 x} \cdot (1 + 9 x)$

Step 4.3.8

Cancel the common factor of $1 + 9 x$ .

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Step 4.3.8.1

Cancel the common factor.

$f (\frac{8 x}{1 + 9 x}) = \frac{x}{1 + 9 x} \cdot (1 + 9 x)$

Step 4.3.8.2

Rewrite the expression.

$f (\frac{8 x}{1 + 9 x}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{8 x}{1 + 9 x}$ is the inverse of $f (x) = \frac{x}{8 - 9 x}$ .

$f^{- 1} (x) = \frac{8 x}{1 + 9 x}$