Trigonometry Examples

$f (x) = 12 x - 5$

Step 1

Write $f (x) = 12 x - 5$ as an equation.

$y = 12 x - 5$

Step 2

Interchange the variables.

$x = 12 y - 5$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $12 y - 5 = x$ .

$12 y - 5 = x$

Step 3.2

Add $5$ to both sides of the equation.

$12 y = x + 5$

Step 3.3

Divide each term in $12 y = x + 5$ by $12$ and simplify.

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Step 3.3.1

Divide each term in $12 y = x + 5$ by $12$ .

$\frac{12 y}{12} = \frac{x}{12} + \frac{5}{12}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $12$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{12 y}{12} = \frac{x}{12} + \frac{5}{12}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{12} + \frac{5}{12}$

Step 4

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

$f^{- 1} (x) = \frac{x}{12} + \frac{5}{12}$

Step 5

Verify if $f^{- 1} (x) = \frac{x}{12} + \frac{5}{12}$ is the inverse of $f (x) = 12 x - 5$ .

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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 $f^{- 1} (12 x - 5)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (12 x - 5) = \frac{12 x - 5}{12} + \frac{5}{12}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (12 x - 5) = \frac{12 x - 5 + 5}{12}$

Step 5.2.4

Combine the opposite terms in $12 x - 5 + 5$ .

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Step 5.2.4.1

Add $- 5$ and $5$ .

$f^{- 1} (12 x - 5) = \frac{12 x + 0}{12}$

Step 5.2.4.2

Add $12 x$ and $0$ .

$f^{- 1} (12 x - 5) = \frac{12 x}{12}$

Step 5.2.5

Cancel the common factor of $12$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (12 x - 5) = \frac{12 x}{12}$

Step 5.2.5.2

Divide $x$ by $1$ .

$f^{- 1} (12 x - 5) = x$

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 $f (\frac{x}{12} + \frac{5}{12})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{12} + \frac{5}{12}) = 12 (\frac{x}{12} + \frac{5}{12}) - 5$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{x}{12} + \frac{5}{12}) = 12 (\frac{x}{12}) + 12 (\frac{5}{12}) - 5$

Step 5.3.3.2

Cancel the common factor of $12$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{x}{12} + \frac{5}{12}) = 12 (\frac{x}{12}) + 12 (\frac{5}{12}) - 5$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{x}{12} + \frac{5}{12}) = x + 12 (\frac{5}{12}) - 5$

Step 5.3.3.3

Cancel the common factor of $12$ .

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Step 5.3.3.3.1

Cancel the common factor.

$f (\frac{x}{12} + \frac{5}{12}) = x + 12 (\frac{5}{12}) - 5$

Step 5.3.3.3.2

Rewrite the expression.

$f (\frac{x}{12} + \frac{5}{12}) = x + 5 - 5$

Step 5.3.4

Combine the opposite terms in $x + 5 - 5$ .

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Step 5.3.4.1

Subtract $5$ from $5$ .

$f (\frac{x}{12} + \frac{5}{12}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{x}{12} + \frac{5}{12}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{12} + \frac{5}{12}$ is the inverse of $f (x) = 12 x - 5$ .

$f^{- 1} (x) = \frac{x}{12} + \frac{5}{12}$