Trigonometry Examples

$\frac{8}{x + 8}$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

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.

$y + 8, 1$

Step 2.2.2

Remove parentheses.

$y + 8, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$y + 8$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $y + 8$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Rewrite the expression.

$8 = x (y + 8)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$8 = x y + x \cdot 8$

Step 2.3.3.2

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

$8 = x y + 8 x$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $x y + 8 x = 8$ .

$x y + 8 x = 8$

Step 2.4.2

Subtract $8 x$ from both sides of the equation.

$x y = 8 - 8 x$

Step 2.4.3

Divide each term in $x y = 8 - 8 x$ by $x$ and simplify.

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Step 2.4.3.1

Divide each term in $x y = 8 - 8 x$ by $x$ .

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

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $x$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

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

Step 2.4.3.2.1.2

Divide $y$ by $1$ .

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

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Cancel the common factor of $x$ .

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Step 2.4.3.3.1.1

Cancel the common factor.

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

Step 2.4.3.3.1.2

Divide $- 8$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.3.2

Cancel the common factor of $8$ .

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Step 4.2.3.2.1

Cancel the common factor.

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

Step 4.2.3.2.2

Rewrite the expression.

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

Step 4.2.4

Combine the opposite terms in $x + 8 - 8$ .

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Step 4.2.4.1

Subtract $8$ from $8$ .

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

Step 4.2.4.2

Add $x$ and $0$ .

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

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

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

Step 4.3.3

Cancel the common factor of $8$ and $(\frac{8}{x} - 8) + 8$ .

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Step 4.3.3.1

Factor $8$ out of $8$ .

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

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

Factor $8$ out of $\frac{8}{x}$ .

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

Step 4.3.3.2.2

Factor $8$ out of $- 8$ .

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

Step 4.3.3.2.3

Factor $8$ out of $8 \frac{1}{x} + 8 \cdot - 1$ .

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

Step 4.3.3.2.4

Factor $8$ out of $8$ .

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

Step 4.3.3.2.5

Factor $8$ out of $8 (\frac{1}{x} - 1) + 8 (1)$ .

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

Step 4.3.3.2.6

Cancel the common factor.

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

Step 4.3.3.2.7

Rewrite the expression.

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

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Add $- 1$ and $1$ .

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

Step 4.3.4.2

Add $\frac{1}{x}$ and $0$ .

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

Step 4.3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.6

Multiply $x$ by $1$ .

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

Step 4.4

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

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