Trigonometry Examples

$\frac{6 x}{x + 5}$

Step 1

Interchange the variables.

$x = \frac{6 y}{y + 5}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{6 y}{y + 5} = x$ .

$\frac{6 y}{y + 5} = x$

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.

$y + 5, 1$

Step 2.2.2

Remove parentheses.

$y + 5, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$y + 5$

Step 2.3

Multiply each term in $\frac{6 y}{y + 5} = x$ by $y + 5$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{6 y}{y + 5} = x$ by $y + 5$ .

$\frac{6 y}{y + 5} (y + 5) = x (y + 5)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $y + 5$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{6 y}{y + 5} (y + 5) = x (y + 5)$

Step 2.3.2.1.2

Rewrite the expression.

$6 y = x (y + 5)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$6 y = x y + x \cdot 5$

Step 2.3.3.2

Move $5$ to the left of $x$ .

$6 y = x y + 5 x$

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $x y$ from both sides of the equation.

$6 y - x y = 5 x$

Step 2.4.2

Factor $y$ out of $6 y - x y$ .

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Step 2.4.2.1

Factor $y$ out of $6 y$ .

$y \cdot 6 - x y = 5 x$

Step 2.4.2.2

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

$y \cdot 6 + y (- x) = 5 x$

Step 2.4.2.3

Factor $y$ out of $y \cdot 6 + y (- x)$ .

$y (6 - x) = 5 x$

Step 2.4.3

Divide each term in $y (6 - x) = 5 x$ by $6 - x$ and simplify.

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Step 2.4.3.1

Divide each term in $y (6 - x) = 5 x$ by $6 - x$ .

$\frac{y (6 - x)}{6 - x} = \frac{5 x}{6 - x}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $6 - x$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{y (6 - x)}{6 - x} = \frac{5 x}{6 - x}$

Step 2.4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 x}{6 - x}$

Step 3

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

$f^{- 1} (x) = \frac{5 x}{6 - x}$

Step 4

Verify if $f^{- 1} (x) = \frac{5 x}{6 - x}$ is the inverse of $f (x) = \frac{6 x}{x + 5}$ .

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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 $f^{- 1} (\frac{6 x}{x + 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{5 (\frac{6 x}{x + 5})}{6 - (\frac{6 x}{x + 5})}$

Step 4.2.3

Combine $5$ and $\frac{6 x}{x + 5}$ .

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{\frac{5 (6 x)}{x + 5}}{6 - \frac{6 x}{x + 5}}$

Step 4.2.4

Simplify the denominator.

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Step 4.2.4.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{\frac{5 (6 x)}{x + 5}}{\frac{6 (x + 5)}{x + 5} - \frac{6 x}{x + 5}}$

Step 4.2.4.2

Combine the numerators over the common denominator.

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{\frac{5 (6 x)}{x + 5}}{\frac{6 (x + 5) - 6 x}{x + 5}}$

Step 4.2.4.3

Reorder terms.

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{\frac{5 (6 x)}{x + 5}}{\frac{- 6 x + 6 (x + 5)}{x + 5}}$

Step 4.2.4.4

Rewrite $\frac{- 6 x + 6 (x + 5)}{x + 5}$ in a factored form.

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Step 4.2.4.4.1

Factor $6$ out of $- 6 x + 6 (x + 5)$ .

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Step 4.2.4.4.1.1

Factor $6$ out of $- 6 x$ .

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{\frac{5 (6 x)}{x + 5}}{\frac{6 (- x) + 6 (x + 5)}{x + 5}}$

Step 4.2.4.4.1.2

Factor $6$ out of $6 (- x) + 6 (x + 5)$ .

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{\frac{5 (6 x)}{x + 5}}{\frac{6 (- x + x + 5)}{x + 5}}$

Step 4.2.4.4.2

Add $- x$ and $x$ .

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{\frac{5 (6 x)}{x + 5}}{\frac{6 (0 + 5)}{x + 5}}$

Step 4.2.4.4.3

Add $0$ and $5$ .

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{\frac{5 (6 x)}{x + 5}}{\frac{6 \cdot 5}{x + 5}}$

Step 4.2.4.5

Multiply $6$ by $5$ .

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{\frac{5 (6 x)}{x + 5}}{\frac{30}{x + 5}}$

Step 4.2.5

Multiply $5$ by $6$ .

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{\frac{30 x}{x + 5}}{\frac{30}{x + 5}}$

Step 4.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{30 x}{x + 5} \cdot \frac{x + 5}{30}$

Step 4.2.7

Cancel the common factor of $30$ .

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Step 4.2.7.1

Factor $30$ out of $30 x$ .

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{30 (x)}{x + 5} \cdot \frac{x + 5}{30}$

Step 4.2.7.2

Cancel the common factor.

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{30 x}{x + 5} \cdot \frac{x + 5}{30}$

Step 4.2.7.3

Rewrite the expression.

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{x}{x + 5} \cdot (x + 5)$

Step 4.2.8

Cancel the common factor of $x + 5$ .

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Step 4.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{6 x}{x + 5}) = \frac{x}{x + 5} \cdot (x + 5)$

Step 4.2.8.2

Rewrite the expression.

$f^{- 1} (\frac{6 x}{x + 5}) = x$

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 $f (\frac{5 x}{6 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x}{6 - x}) = \frac{6 (\frac{5 x}{6 - x})}{(\frac{5 x}{6 - x}) + 5}$

Step 4.3.3

Combine $6$ and $\frac{5 x}{6 - x}$ .

$f (\frac{5 x}{6 - x}) = \frac{\frac{6 (5 x)}{6 - x}}{\frac{5 x}{6 - x} + 5}$

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{6 - x}{6 - x}$ .

$f (\frac{5 x}{6 - x}) = \frac{\frac{6 (5 x)}{6 - x}}{\frac{5 x}{6 - x} + \frac{5 (6 - x)}{6 - x}}$

Step 4.3.4.2

Combine the numerators over the common denominator.

$f (\frac{5 x}{6 - x}) = \frac{\frac{6 (5 x)}{6 - x}}{\frac{5 x + 5 (6 - x)}{6 - x}}$

Step 4.3.4.3

Reorder terms.

$f (\frac{5 x}{6 - x}) = \frac{\frac{6 (5 x)}{6 - x}}{\frac{5 x + 5 (6 - x)}{- x + 6}}$

Step 4.3.4.4

Rewrite $\frac{5 x + 5 (6 - x)}{- x + 6}$ in a factored form.

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Step 4.3.4.4.1

Factor $5$ out of $5 x + 5 (6 - x)$ .

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Step 4.3.4.4.1.1

Factor $5$ out of $5 x$ .

$f (\frac{5 x}{6 - x}) = \frac{\frac{6 (5 x)}{6 - x}}{\frac{5 (x) + 5 (6 - x)}{- x + 6}}$

Step 4.3.4.4.1.2

Factor $5$ out of $5 (x) + 5 (6 - x)$ .

$f (\frac{5 x}{6 - x}) = \frac{\frac{6 (5 x)}{6 - x}}{\frac{5 (x + 6 - x)}{- x + 6}}$

Step 4.3.4.4.2

Subtract $x$ from $x$ .

$f (\frac{5 x}{6 - x}) = \frac{\frac{6 (5 x)}{6 - x}}{\frac{5 (0 + 6)}{- x + 6}}$

Step 4.3.4.4.3

Add $0$ and $6$ .

$f (\frac{5 x}{6 - x}) = \frac{\frac{6 (5 x)}{6 - x}}{\frac{5 \cdot 6}{- x + 6}}$

Step 4.3.4.5

Multiply $5$ by $6$ .

$f (\frac{5 x}{6 - x}) = \frac{\frac{6 (5 x)}{6 - x}}{\frac{30}{- x + 6}}$

Step 4.3.5

Multiply $6$ by $5$ .

$f (\frac{5 x}{6 - x}) = \frac{\frac{30 x}{6 - x}}{\frac{30}{- x + 6}}$

Step 4.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5 x}{6 - x}) = \frac{30 x}{6 - x} \cdot \frac{- x + 6}{30}$

Step 4.3.7

Cancel the common factor of $30$ .

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Step 4.3.7.1

Factor $30$ out of $30 x$ .

$f (\frac{5 x}{6 - x}) = \frac{30 (x)}{6 - x} \cdot \frac{- x + 6}{30}$

Step 4.3.7.2

Cancel the common factor.

$f (\frac{5 x}{6 - x}) = \frac{30 x}{6 - x} \cdot \frac{- x + 6}{30}$

Step 4.3.7.3

Rewrite the expression.

$f (\frac{5 x}{6 - x}) = \frac{x}{6 - x} \cdot (- x + 6)$

Step 4.3.8

Multiply $\frac{x}{6 - x}$ by $- x + 6$ .

$f (\frac{5 x}{6 - x}) = \frac{x (- x + 6)}{6 - x}$

Step 4.3.9

Cancel the common factor of $- x + 6$ and $6 - x$ .

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Step 4.3.9.1

Reorder terms.

$f (\frac{5 x}{6 - x}) = \frac{x (- x + 6)}{- x + 6}$

Step 4.3.9.2

Cancel the common factor.

$f (\frac{5 x}{6 - x}) = \frac{x (- x + 6)}{- x + 6}$

Step 4.3.9.3

Divide $x$ by $1$ .

$f (\frac{5 x}{6 - x}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x}{6 - x}$ is the inverse of $f (x) = \frac{6 x}{x + 5}$ .

$f^{- 1} (x) = \frac{5 x}{6 - x}$