Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Multiply the equation by $8 y - 9$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Apply the distributive property.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Cancel the common factor.

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

Step 2.3.1.2

Rewrite the expression.

$8 y x - 9 x = y$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Subtract $y$ from both sides of the equation.

$8 y x - 9 x - y = 0$

Step 2.4.2

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

$8 y x - y = 9 x$

Step 2.4.3

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

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Step 2.4.3.1

Factor $y$ out of $8 y x$ .

$y (8 x) - y = 9 x$

Step 2.4.3.2

Factor $y$ out of $- y$ .

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

Step 2.4.3.3

Factor $y$ out of $y (8 x) + y \cdot - 1$ .

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

Step 2.4.4

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

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Step 2.4.4.1

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

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

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $8 x - 1$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

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

Step 2.4.4.2.1.2

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

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

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

Step 4.2.4

Simplify the denominator.

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Step 4.2.4.1

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

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

Step 4.2.4.2

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

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

Step 4.2.4.3

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

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

Step 4.2.4.4

Combine the numerators over the common denominator.

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

Step 4.2.4.5

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

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Step 4.2.4.5.1

Apply the distributive property.

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

Step 4.2.4.5.2

Multiply $8$ by $- 1$ .

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

Step 4.2.4.5.3

Multiply $- 1$ by $- 9$ .

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

Step 4.2.4.5.4

Subtract $8 x$ from $8 x$ .

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

Step 4.2.4.5.5

Add $0$ and $9$ .

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

Step 4.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.6

Cancel the common factor of $9$ .

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Step 4.2.6.1

Factor $9$ out of $9 x$ .

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

Step 4.2.6.2

Cancel the common factor.

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

Step 4.2.6.3

Rewrite the expression.

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

Step 4.2.7

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

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Step 4.2.7.1

Cancel the common factor.

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

Step 4.2.7.2

Rewrite the expression.

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

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

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

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

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

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Step 4.3.4.1.1

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

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

Step 4.3.4.1.2

Multiply $9$ by $8$ .

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

Step 4.3.4.2

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

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

Step 4.3.4.3

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

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

Step 4.3.4.4

Combine the numerators over the common denominator.

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

Step 4.3.4.5

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

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Step 4.3.4.5.1

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

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Step 4.3.4.5.1.1

Factor $9$ out of $72 x$ .

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

Step 4.3.4.5.1.2

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

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

Step 4.3.4.5.1.3

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

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

Step 4.3.4.5.2

Apply the distributive property.

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

Step 4.3.4.5.3

Multiply $8$ by $- 1$ .

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

Step 4.3.4.5.4

Multiply $- 1$ by $- 1$ .

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

Step 4.3.4.5.5

Subtract $8 x$ from $8 x$ .

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

Step 4.3.4.5.6

Add $0$ and $1$ .

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

Step 4.3.4.6

Multiply $9$ by $1$ .

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

Step 4.3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.6

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

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

Step 4.3.7

Cancel the common factor of $9$ .

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Step 4.3.7.1

Factor $9$ out of $9 x$ .

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

Step 4.3.7.2

Cancel the common factor.

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

Step 4.3.7.3

Rewrite the expression.

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

Step 4.3.8

Cancel the common factor of $8 x - 1$ .

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Step 4.3.8.1

Cancel the common factor.

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

Step 4.3.8.2

Rewrite the expression.

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

Step 4.4

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

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