Trigonometry Examples

$f (x) = \frac{17}{6} x - 19$

Step 1

Write $f (x) = \frac{17}{6} x - 19$ as an equation.

$y = \frac{17}{6} x - 19$

Step 2

Interchange the variables.

$x = \frac{17}{6} y - 19$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{17}{6} y - 19 = x$ .

$\frac{17}{6} y - 19 = x$

Step 3.2

Combine $\frac{17}{6}$ and $y$ .

$\frac{17 y}{6} - 19 = x$

Step 3.3

Add $19$ to both sides of the equation.

$\frac{17 y}{6} = x + 19$

Step 3.4

Multiply both sides of the equation by $\frac{6}{17}$ .

$\frac{6}{17} \cdot \frac{17 y}{6} = \frac{6}{17} (x + 19)$

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify $\frac{6}{17} \cdot \frac{17 y}{6}$ .

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Step 3.5.1.1.1

Cancel the common factor of $6$ .

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Step 3.5.1.1.1.1

Cancel the common factor.

$\frac{6}{17} \cdot \frac{17 y}{6} = \frac{6}{17} (x + 19)$

Step 3.5.1.1.1.2

Rewrite the expression.

$\frac{1}{17} (17 y) = \frac{6}{17} (x + 19)$

Step 3.5.1.1.2

Cancel the common factor of $17$ .

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Step 3.5.1.1.2.1

Factor $17$ out of $17 y$ .

$\frac{1}{17} (17 (y)) = \frac{6}{17} (x + 19)$

Step 3.5.1.1.2.2

Cancel the common factor.

$\frac{1}{17} (17 y) = \frac{6}{17} (x + 19)$

Step 3.5.1.1.2.3

Rewrite the expression.

$y = \frac{6}{17} (x + 19)$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $\frac{6}{17} (x + 19)$ .

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Step 3.5.2.1.1

Apply the distributive property.

$y = \frac{6}{17} x + \frac{6}{17} \cdot 19$

Step 3.5.2.1.2

Combine $\frac{6}{17}$ and $x$ .

$y = \frac{6 x}{17} + \frac{6}{17} \cdot 19$

Step 3.5.2.1.3

Multiply $\frac{6}{17} \cdot 19$ .

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Step 3.5.2.1.3.1

Combine $\frac{6}{17}$ and $19$ .

$y = \frac{6 x}{17} + \frac{6 \cdot 19}{17}$

Step 3.5.2.1.3.2

Multiply $6$ by $19$ .

$y = \frac{6 x}{17} + \frac{114}{17}$

Step 4

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

$f^{- 1} (x) = \frac{6 x}{17} + \frac{114}{17}$

Step 5

Verify if $f^{- 1} (x) = \frac{6 x}{17} + \frac{114}{17}$ is the inverse of $f (x) = \frac{17}{6} x - 19$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (\frac{17}{6} x - 19)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{17}{6} x - 19) = \frac{6 (\frac{17}{6} x - 19)}{17} + \frac{114}{17}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{17}{6} x - 19) = \frac{6 (\frac{17}{6} x - 19) + 114}{17}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Combine $\frac{17}{6}$ and $x$ .

$f^{- 1} (\frac{17}{6} x - 19) = \frac{6 (\frac{17 x}{6} - 19) + 114}{17}$

Step 5.2.4.2

Apply the distributive property.

$f^{- 1} (\frac{17}{6} x - 19) = \frac{6 (\frac{17 x}{6}) + 6 \cdot - 19 + 114}{17}$

Step 5.2.4.3

Cancel the common factor of $6$ .

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Step 5.2.4.3.1

Cancel the common factor.

$f^{- 1} (\frac{17}{6} x - 19) = \frac{6 (\frac{17 x}{6}) + 6 \cdot - 19 + 114}{17}$

Step 5.2.4.3.2

Rewrite the expression.

$f^{- 1} (\frac{17}{6} x - 19) = \frac{17 x + 6 \cdot - 19 + 114}{17}$

Step 5.2.4.4

Multiply $6$ by $- 19$ .

$f^{- 1} (\frac{17}{6} x - 19) = \frac{17 x - 114 + 114}{17}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $17 x - 114 + 114$ .

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Step 5.2.5.1.1

Add $- 114$ and $114$ .

$f^{- 1} (\frac{17}{6} x - 19) = \frac{17 x + 0}{17}$

Step 5.2.5.1.2

Add $17 x$ and $0$ .

$f^{- 1} (\frac{17}{6} x - 19) = \frac{17 x}{17}$

Step 5.2.5.2

Cancel the common factor of $17$ .

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Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{17}{6} x - 19) = \frac{17 x}{17}$

Step 5.2.5.2.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{17}{6} x - 19) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (\frac{6 x}{17} + \frac{114}{17})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6 x}{17} + \frac{114}{17}) = \frac{17}{6} \cdot (\frac{6 x}{17} + \frac{114}{17}) - 19$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{6 x}{17} + \frac{114}{17}) = \frac{17}{6} \cdot \frac{6 x}{17} + \frac{17}{6} \cdot \frac{114}{17} - 19$

Step 5.3.3.2

Cancel the common factor of $17$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{6 x}{17} + \frac{114}{17}) = \frac{17}{6} \cdot \frac{6 x}{17} + \frac{17}{6} \cdot \frac{114}{17} - 19$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{6 x}{17} + \frac{114}{17}) = \frac{1}{6} \cdot (6 x) + \frac{17}{6} \cdot \frac{114}{17} - 19$

Step 5.3.3.3

Cancel the common factor of $6$ .

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Step 5.3.3.3.1

Factor $6$ out of $6 x$ .

$f (\frac{6 x}{17} + \frac{114}{17}) = \frac{1}{6} \cdot (6 (x)) + \frac{17}{6} \cdot \frac{114}{17} - 19$

Step 5.3.3.3.2

Cancel the common factor.

$f (\frac{6 x}{17} + \frac{114}{17}) = \frac{1}{6} \cdot (6 x) + \frac{17}{6} \cdot \frac{114}{17} - 19$

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{6 x}{17} + \frac{114}{17}) = x + \frac{17}{6} \cdot \frac{114}{17} - 19$

Step 5.3.3.4

Cancel the common factor of $17$ .

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Step 5.3.3.4.1

Cancel the common factor.

$f (\frac{6 x}{17} + \frac{114}{17}) = x + \frac{17}{6} \cdot \frac{114}{17} - 19$

Step 5.3.3.4.2

Rewrite the expression.

$f (\frac{6 x}{17} + \frac{114}{17}) = x + \frac{1}{6} \cdot 114 - 19$

Step 5.3.3.5

Cancel the common factor of $6$ .

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Step 5.3.3.5.1

Factor $6$ out of $114$ .

$f (\frac{6 x}{17} + \frac{114}{17}) = x + \frac{1}{6} \cdot (6 (19)) - 19$

Step 5.3.3.5.2

Cancel the common factor.

$f (\frac{6 x}{17} + \frac{114}{17}) = x + \frac{1}{6} \cdot (6 \cdot 19) - 19$

Step 5.3.3.5.3

Rewrite the expression.

$f (\frac{6 x}{17} + \frac{114}{17}) = x + 19 - 19$

Step 5.3.4

Combine the opposite terms in $x + 19 - 19$ .

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Step 5.3.4.1

Subtract $19$ from $19$ .

$f (\frac{6 x}{17} + \frac{114}{17}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{6 x}{17} + \frac{114}{17}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6 x}{17} + \frac{114}{17}$ is the inverse of $f (x) = \frac{17}{6} x - 19$ .

$f^{- 1} (x) = \frac{6 x}{17} + \frac{114}{17}$