Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

Add $6$ to both sides of the equation.

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

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.

$y, 1, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$y$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $y$ .

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Step 2.4.2.1.1

Cancel the common factor.

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

Step 2.4.2.1.2

Rewrite the expression.

$5 = x y + 6 y$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $x y + 6 y = 5$ .

$x y + 6 y = 5$

Step 2.5.2

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

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Step 2.5.2.1

Factor $y$ out of $x y$ .

$y x + 6 y = 5$

Step 2.5.2.2

Factor $y$ out of $6 y$ .

$y x + y \cdot 6 = 5$

Step 2.5.2.3

Factor $y$ out of $y x + y \cdot 6$ .

$y (x + 6) = 5$

Step 2.5.3

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

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Step 2.5.3.1

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

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

Step 2.5.3.2

Simplify the left side.

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Step 2.5.3.2.1

Cancel the common factor of $x + 6$ .

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Step 2.5.3.2.1.1

Cancel the common factor.

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

Step 2.5.3.2.1.2

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Simplify the denominator.

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Step 4.2.3.1

Add $- 6$ and $6$ .

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

Step 4.2.3.2

Add $\frac{5}{x}$ and $0$ .

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

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.5

Cancel the common factor of $5$ .

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Step 4.2.5.1

Cancel the common factor.

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

Step 4.2.5.2

Rewrite the expression.

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

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

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.3.2

Cancel the common factor of $5$ .

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Step 4.3.3.2.1

Cancel the common factor.

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

Step 4.3.3.2.2

Rewrite the expression.

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

Step 4.3.4

Combine the opposite terms in $x + 6 - 6$ .

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Step 4.3.4.1

Subtract $6$ from $6$ .

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

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

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

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