Trigonometry Examples

$f (x) = 3 x - 0.5$

Step 1

Write $f (x) = 3 x - 0.5$ as an equation.

$y = 3 x - 0.5$

Step 2

Interchange the variables.

$x = 3 y - 0.5$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 y - 0.5 = x$ .

$3 y - 0.5 = x$

Step 3.2

Add $0.5$ to both sides of the equation.

$3 y = x + 0.5$

Step 3.3

Divide each term in $3 y = x + 0.5$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 y = x + 0.5$ by $3$ .

$\frac{3 y}{3} = \frac{x}{3} + \frac{0.5}{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{x}{3} + \frac{0.5}{3}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{3} + \frac{0.5}{3}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0.5$ by $3$ .

$y = \frac{x}{3} + 0.1\overline{6}$

Step 4

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

$f^{- 1} (x) = \frac{x}{3} + 0.1\overline{6}$

Step 5

Verify if $f^{- 1} (x) = \frac{x}{3} + 0.1\overline{6}$ is the inverse of $f (x) = 3 x - 0.5$ .

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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 $f^{- 1} (3 x - 0.5)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 x - 0.5) = \frac{3 x - 0.5}{3} + 0.1\overline{6}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Factor $0.5$ out of $3 x - 0.5$ .

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Step 5.2.3.1.1

Factor $0.5$ out of $3 x$ .

$f^{- 1} (3 x - 0.5) = \frac{0.5 (6 x) - 0.5}{3} + 0.1\overline{6}$

Step 5.2.3.1.2

Factor $0.5$ out of $- 0.5$ .

$f^{- 1} (3 x - 0.5) = \frac{0.5 (6 x) + 0.5 (- 1)}{3} + 0.1\overline{6}$

Step 5.2.3.1.3

Factor $0.5$ out of $0.5 (6 x) + 0.5 (- 1)$ .

$f^{- 1} (3 x - 0.5) = \frac{0.5 (6 x - 1)}{3} + 0.1\overline{6}$

Step 5.2.3.2

Factor $3$ out of $3$ .

$f^{- 1} (3 x - 0.5) = \frac{0.5 (6 x - 1)}{3 (1)} + 0.1\overline{6}$

Step 5.2.3.3

Separate fractions.

$f^{- 1} (3 x - 0.5) = \frac{0.5}{3} \cdot \frac{6 x - 1}{1} + 0.1\overline{6}$

Step 5.2.3.4

Divide $0.5$ by $3$ .

$f^{- 1} (3 x - 0.5) = 0.1\overline{6} (\frac{6 x - 1}{1}) + 0.1\overline{6}$

Step 5.2.3.5

Divide $6 x - 1$ by $1$ .

$f^{- 1} (3 x - 0.5) = 0.1\overline{6} (6 x - 1) + 0.1\overline{6}$

Step 5.2.3.6

Apply the distributive property.

$f^{- 1} (3 x - 0.5) = 0.1\overline{6} (6 x) + 0.1\overline{6} \cdot - 1 + 0.1\overline{6}$

Step 5.2.3.7

Multiply $6$ by $0.1\overline{6}$ .

$f^{- 1} (3 x - 0.5) = 1 x + 0.1\overline{6} \cdot - 1 + 0.1\overline{6}$

Step 5.2.3.8

Multiply $0.1\overline{6}$ by $- 1$ .

$f^{- 1} (3 x - 0.5) = 1 x - 0.1\overline{6} + 0.1\overline{6}$

Step 5.2.4

Combine the opposite terms in $1 x - 0.1\overline{6} + 0.1\overline{6}$ .

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Step 5.2.4.1

Add $- 0.1\overline{6}$ and $0.1\overline{6}$ .

$f^{- 1} (3 x - 0.5) = 1 x + 0$

Step 5.2.4.2

Add $1 x$ and $0$ .

$f^{- 1} (3 x - 0.5) = 1 x$

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 $f (\frac{x}{3} + 0.1\overline{6})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{3} + 0.1\overline{6}) = 3 (\frac{x}{3} + 0.1\overline{6}) - 0.5$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{x}{3} + 0.1\overline{6}) = 3 (\frac{x}{3}) + 3 \cdot 0.1\overline{6} - 0.5$

Step 5.3.3.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{x}{3} + 0.1\overline{6}) = 3 (\frac{x}{3}) + 3 \cdot 0.1\overline{6} - 0.5$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{x}{3} + 0.1\overline{6}) = x + 3 \cdot 0.1\overline{6} - 0.5$

Step 5.3.3.3

Multiply $3$ by $0.1\overline{6}$ .

$f (\frac{x}{3} + 0.1\overline{6}) = x + 0.5 - 0.5$

Step 5.3.4

Subtract $0.5$ from $0.5$ .

$f (\frac{x}{3} + 0.1\overline{6}) = x + 0$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{3} + 0.1\overline{6}$ is the inverse of $f (x) = 3 x - 0.5$ .

$f^{- 1} (x) = \frac{x}{3} + 0.1\overline{6}$