Trigonometry Examples

$4 x + 2 - 6 y = 0$

Step 1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1

Subtract $4 x$ from both sides of the equation.

$2 - 6 y = - 4 x$

Step 1.2

Subtract $2$ from both sides of the equation.

$- 6 y = - 4 x - 2$

Step 2

Divide each term in $- 6 y = - 4 x - 2$ by $- 6$ and simplify.

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Step 2.1

Divide each term in $- 6 y = - 4 x - 2$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{- 4 x}{- 6} + \frac{- 2}{- 6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 6$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{- 4 x}{- 6} + \frac{- 2}{- 6}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4 x}{- 6} + \frac{- 2}{- 6}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Cancel the common factor of $- 4$ and $- 6$ .

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Step 2.3.1.1.1

Factor $- 2$ out of $- 4 x$ .

$y = \frac{- 2 (2 x)}{- 6} + \frac{- 2}{- 6}$

Step 2.3.1.1.2

Cancel the common factors.

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Step 2.3.1.1.2.1

Factor $- 2$ out of $- 6$ .

$y = \frac{- 2 (2 x)}{- 2 (3)} + \frac{- 2}{- 6}$

Step 2.3.1.1.2.2

Cancel the common factor.

$y = \frac{- 2 (2 x)}{- 2 \cdot 3} + \frac{- 2}{- 6}$

Step 2.3.1.1.2.3

Rewrite the expression.

$y = \frac{2 x}{3} + \frac{- 2}{- 6}$

Step 2.3.1.2

Cancel the common factor of $- 2$ and $- 6$ .

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Step 2.3.1.2.1

Factor $- 2$ out of $- 2$ .

$y = \frac{2 x}{3} + \frac{- 2 (1)}{- 6}$

Step 2.3.1.2.2

Cancel the common factors.

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Step 2.3.1.2.2.1

Factor $- 2$ out of $- 6$ .

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

Step 2.3.1.2.2.2

Cancel the common factor.

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

Step 2.3.1.2.2.3

Rewrite the expression.

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

Step 3

Interchange the variables.

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

Step 4

Solve for $y$ .

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Step 4.1

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

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

Step 4.2

Subtract $\frac{1}{3}$ from both sides of the equation.

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

Step 4.3

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

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

Step 4.4

Simplify both sides of the equation.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

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

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Step 4.4.1.1.1

Cancel the common factor of $3$ .

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Step 4.4.1.1.1.1

Cancel the common factor.

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

Step 4.4.1.1.1.2

Rewrite the expression.

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

Step 4.4.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.1.1.2.1

Factor $2$ out of $2 y$ .

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

Step 4.4.1.1.2.2

Cancel the common factor.

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

Step 4.4.1.1.2.3

Rewrite the expression.

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

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

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

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Step 4.4.2.1.1

Apply the distributive property.

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

Step 4.4.2.1.2

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

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

Step 4.4.2.1.3

Cancel the common factor of $3$ .

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Step 4.4.2.1.3.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 4.4.2.1.3.2

Cancel the common factor.

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

Step 4.4.2.1.3.3

Rewrite the expression.

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

Step 4.4.2.1.4

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 4.4.2.1.5

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

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

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Step 6.1

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

Step 6.2

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

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Step 6.2.1

Set up the composite result function.

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

Step 6.2.2

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

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

Step 6.2.3

Combine the numerators over the common denominator.

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

Step 6.2.4

Simplify each term.

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Step 6.2.4.1

Apply the distributive property.

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

Step 6.2.4.2

Cancel the common factor of $3$ .

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Step 6.2.4.2.1

Cancel the common factor.

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

Step 6.2.4.2.2

Rewrite the expression.

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

Step 6.2.4.3

Cancel the common factor of $3$ .

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Step 6.2.4.3.1

Cancel the common factor.

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

Step 6.2.4.3.2

Rewrite the expression.

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

Step 6.2.5

Simplify terms.

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Step 6.2.5.1

Combine the opposite terms in $2 x + 1 - 1$ .

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Step 6.2.5.1.1

Subtract $1$ from $1$ .

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

Step 6.2.5.1.2

Add $2 x$ and $0$ .

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

Step 6.2.5.2

Cancel the common factor of $2$ .

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Step 6.2.5.2.1

Cancel the common factor.

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

Step 6.2.5.2.2

Divide $x$ by $1$ .

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

Step 6.3

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

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Step 6.3.1

Set up the composite result function.

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

Step 6.3.2

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

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

Step 6.3.3

Combine the numerators over the common denominator.

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

Step 6.3.4

Simplify each term.

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Step 6.3.4.1

Apply the distributive property.

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

Step 6.3.4.2

Cancel the common factor of $2$ .

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Step 6.3.4.2.1

Cancel the common factor.

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

Step 6.3.4.2.2

Rewrite the expression.

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

Step 6.3.4.3

Cancel the common factor of $2$ .

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Step 6.3.4.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 6.3.4.3.2

Cancel the common factor.

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

Step 6.3.4.3.3

Rewrite the expression.

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

Step 6.3.5

Simplify terms.

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Step 6.3.5.1

Combine the opposite terms in $3 x - 1 + 1$ .

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Step 6.3.5.1.1

Add $- 1$ and $1$ .

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

Step 6.3.5.1.2

Add $3 x$ and $0$ .

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

Step 6.3.5.2

Cancel the common factor of $3$ .

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Step 6.3.5.2.1

Cancel the common factor.

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

Step 6.3.5.2.2

Divide $x$ by $1$ .

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

Step 6.4

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

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