Trigonometry Examples

$\frac{2}{3} x - \frac{1}{2}$

Step 1

Interchange the variables.

$x = \frac{2}{3} y - \frac{1}{2}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{2}{3} y - \frac{1}{2} = x$ .

$\frac{2}{3} y - \frac{1}{2} = x$

Step 2.2

Combine $\frac{2}{3}$ and $y$ .

$\frac{2 y}{3} - \frac{1}{2} = x$

Step 2.3

Add $\frac{1}{2}$ to both sides of the equation.

$\frac{2 y}{3} = x + \frac{1}{2}$

Step 2.4

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 y}{3} = \frac{3}{2} (x + \frac{1}{2})$

Step 2.5

Simplify both sides of the equation.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Simplify $\frac{3}{2} \cdot \frac{2 y}{3}$ .

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Step 2.5.1.1.1

Cancel the common factor of $3$ .

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Step 2.5.1.1.1.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 y}{3} = \frac{3}{2} (x + \frac{1}{2})$

Step 2.5.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 y) = \frac{3}{2} (x + \frac{1}{2})$

Step 2.5.1.1.2

Cancel the common factor of $2$ .

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Step 2.5.1.1.2.1

Factor $2$ out of $2 y$ .

$\frac{1}{2} (2 (y)) = \frac{3}{2} (x + \frac{1}{2})$

Step 2.5.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 y) = \frac{3}{2} (x + \frac{1}{2})$

Step 2.5.1.1.2.3

Rewrite the expression.

$y = \frac{3}{2} (x + \frac{1}{2})$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify $\frac{3}{2} (x + \frac{1}{2})$ .

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Step 2.5.2.1.1

Apply the distributive property.

$y = \frac{3}{2} x + \frac{3}{2} \cdot \frac{1}{2}$

Step 2.5.2.1.2

Combine $\frac{3}{2}$ and $x$ .

$y = \frac{3 x}{2} + \frac{3}{2} \cdot \frac{1}{2}$

Step 2.5.2.1.3

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

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Step 2.5.2.1.3.1

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$y = \frac{3 x}{2} + \frac{3}{2 \cdot 2}$

Step 2.5.2.1.3.2

Multiply $2$ by $2$ .

$y = \frac{3 x}{2} + \frac{3}{4}$

Step 3

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

$f^{- 1} (x) = \frac{3 x}{2} + \frac{3}{4}$

Step 4

Verify if $f^{- 1} (x) = \frac{3 x}{2} + \frac{3}{4}$ is the inverse of $f (x) = \frac{2}{3} x - \frac{1}{2}$ .

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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 $f^{- 1} (\frac{2}{3} x - \frac{1}{2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{3 (\frac{2}{3} x - \frac{1}{2})}{2} + \frac{3}{4}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

Combine $\frac{2}{3}$ and $x$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{3 (\frac{2 x}{3} - \frac{1}{2})}{2} + \frac{3}{4}$

Step 4.2.3.1.2

To write $\frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{3 (\frac{2 x}{3} \cdot \frac{2}{2} - \frac{1}{2})}{2} + \frac{3}{4}$

Step 4.2.3.1.3

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{3 (\frac{2 x}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3})}{2} + \frac{3}{4}$

Step 4.2.3.1.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.3.1.4.1

Multiply $\frac{2 x}{3}$ by $\frac{2}{2}$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{3 (\frac{2 x \cdot 2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3})}{2} + \frac{3}{4}$

Step 4.2.3.1.4.2

Multiply $3$ by $2$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{3 (\frac{2 x \cdot 2}{6} - \frac{1}{2} \cdot \frac{3}{3})}{2} + \frac{3}{4}$

Step 4.2.3.1.4.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{3 (\frac{2 x \cdot 2}{6} - \frac{3}{2 \cdot 3})}{2} + \frac{3}{4}$

Step 4.2.3.1.4.4

Multiply $2$ by $3$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{3 (\frac{2 x \cdot 2}{6} - \frac{3}{6})}{2} + \frac{3}{4}$

Step 4.2.3.1.5

Combine the numerators over the common denominator.

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{3 (\frac{2 x \cdot 2 - 3}{6})}{2} + \frac{3}{4}$

Step 4.2.3.1.6

Multiply $2$ by $2$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{3 (\frac{4 x - 3}{6})}{2} + \frac{3}{4}$

Step 4.2.3.2

Combine $3$ and $\frac{4 x - 3}{6}$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{\frac{3 (4 x - 3)}{6}}{2} + \frac{3}{4}$

Step 4.2.3.3

Reduce the expression $\frac{3 (4 x - 3)}{6}$ by cancelling the common factors.

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Step 4.2.3.3.1

Factor $3$ out of $6$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{\frac{3 (4 x - 3)}{3 \cdot 2}}{2} + \frac{3}{4}$

Step 4.2.3.3.2

Cancel the common factor.

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{\frac{3 (4 x - 3)}{3 \cdot 2}}{2} + \frac{3}{4}$

Step 4.2.3.3.3

Rewrite the expression.

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{\frac{4 x - 3}{2}}{2} + \frac{3}{4}$

Step 4.2.3.4

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{4 x - 3}{2} \cdot \frac{1}{2} + \frac{3}{4}$

Step 4.2.3.5

Multiply $\frac{4 x - 3}{2} \cdot \frac{1}{2}$ .

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Step 4.2.3.5.1

Multiply $\frac{4 x - 3}{2}$ by $\frac{1}{2}$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{4 x - 3}{2 \cdot 2} + \frac{3}{4}$

Step 4.2.3.5.2

Multiply $2$ by $2$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{4 x - 3}{4} + \frac{3}{4}$

Step 4.2.4

Simplify terms.

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Step 4.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{4 x - 3 + 3}{4}$

Step 4.2.4.2

Combine the opposite terms in $4 x - 3 + 3$ .

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Step 4.2.4.2.1

Add $- 3$ and $3$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{4 x + 0}{4}$

Step 4.2.4.2.2

Add $4 x$ and $0$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{4 x}{4}$

Step 4.2.4.3

Cancel the common factor of $4$ .

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Step 4.2.4.3.1

Cancel the common factor.

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = \frac{4 x}{4}$

Step 4.2.4.3.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{2}{3} x - \frac{1}{2}) = x$

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 $f (\frac{3 x}{2} + \frac{3}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x}{2} + \frac{3}{4}) = \frac{2}{3} \cdot (\frac{3 x}{2} + \frac{3}{4}) - \frac{1}{2}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (\frac{3 x}{2} + \frac{3}{4}) = \frac{2}{3} \cdot \frac{3 x}{2} + \frac{2}{3} \cdot \frac{3}{4} - \frac{1}{2}$

Step 4.3.3.2

Cancel the common factor of $2$ .

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Step 4.3.3.2.1

Cancel the common factor.

$f (\frac{3 x}{2} + \frac{3}{4}) = \frac{2}{3} \cdot \frac{3 x}{2} + \frac{2}{3} \cdot \frac{3}{4} - \frac{1}{2}$

Step 4.3.3.2.2

Rewrite the expression.

$f (\frac{3 x}{2} + \frac{3}{4}) = \frac{1}{3} \cdot (3 x) + \frac{2}{3} \cdot \frac{3}{4} - \frac{1}{2}$

Step 4.3.3.3

Cancel the common factor of $3$ .

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Step 4.3.3.3.1

Factor $3$ out of $3 x$ .

$f (\frac{3 x}{2} + \frac{3}{4}) = \frac{1}{3} \cdot (3 (x)) + \frac{2}{3} \cdot \frac{3}{4} - \frac{1}{2}$

Step 4.3.3.3.2

Cancel the common factor.

$f (\frac{3 x}{2} + \frac{3}{4}) = \frac{1}{3} \cdot (3 x) + \frac{2}{3} \cdot \frac{3}{4} - \frac{1}{2}$

Step 4.3.3.3.3

Rewrite the expression.

$f (\frac{3 x}{2} + \frac{3}{4}) = x + \frac{2}{3} \cdot \frac{3}{4} - \frac{1}{2}$

Step 4.3.3.4

Cancel the common factor of $2$ .

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Step 4.3.3.4.1

Factor $2$ out of $4$ .

$f (\frac{3 x}{2} + \frac{3}{4}) = x + \frac{2}{3} \cdot \frac{3}{2 (2)} - \frac{1}{2}$

Step 4.3.3.4.2

Cancel the common factor.

$f (\frac{3 x}{2} + \frac{3}{4}) = x + \frac{2}{3} \cdot \frac{3}{2 \cdot 2} - \frac{1}{2}$

Step 4.3.3.4.3

Rewrite the expression.

$f (\frac{3 x}{2} + \frac{3}{4}) = x + \frac{1}{3} \cdot \frac{3}{2} - \frac{1}{2}$

Step 4.3.3.5

Cancel the common factor of $3$ .

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Step 4.3.3.5.1

Cancel the common factor.

$f (\frac{3 x}{2} + \frac{3}{4}) = x + \frac{1}{3} \cdot \frac{3}{2} - \frac{1}{2}$

Step 4.3.3.5.2

Rewrite the expression.

$f (\frac{3 x}{2} + \frac{3}{4}) = x + \frac{1}{2} - \frac{1}{2}$

Step 4.3.4

Combine the opposite terms in $x + \frac{1}{2} - \frac{1}{2}$ .

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Step 4.3.4.1

Combine the numerators over the common denominator.

$f (\frac{3 x}{2} + \frac{3}{4}) = x + \frac{1 - 1}{2}$

Step 4.3.4.2

Subtract $1$ from $1$ .

$f (\frac{3 x}{2} + \frac{3}{4}) = x + \frac{0}{2}$

Step 4.3.4.3

Divide $0$ by $2$ .

$f (\frac{3 x}{2} + \frac{3}{4}) = x + 0$

Step 4.3.4.4

Add $x$ and $0$ .

$f (\frac{3 x}{2} + \frac{3}{4}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{2} + \frac{3}{4}$ is the inverse of $f (x) = \frac{2}{3} x - \frac{1}{2}$ .

$f^{- 1} (x) = \frac{3 x}{2} + \frac{3}{4}$