Trigonometry Examples

$2 y = 6 x + 3$

Step 1

Divide each term in $2 y = 6 x + 3$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 y = 6 x + 3$ by $2$ .

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Divide $y$ by $1$ .

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

Step 1.3

Simplify the right side.

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Step 1.3.1

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

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Step 1.3.1.1

Factor $2$ out of $6 x$ .

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

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $2$ .

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

Step 1.3.1.2.2

Cancel the common factor.

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

Step 1.3.1.2.3

Rewrite the expression.

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

Step 1.3.1.2.4

Divide $3 x$ by $1$ .

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Subtract $\frac{3}{2}$ from both sides of the equation.

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

Step 3.3

Divide each term in $3 y = x - \frac{3}{2}$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 y = x - \frac{3}{2}$ by $3$ .

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

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.

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

Step 3.3.2.1.2

Divide $y$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3.3.1.2

Cancel the common factor of $3$ .

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Step 3.3.3.1.2.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 3.3.3.1.2.2

Factor $3$ out of $- 3$ .

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

Step 3.3.3.1.2.3

Cancel the common factor.

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

Step 3.3.3.1.2.4

Rewrite the expression.

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

Step 3.3.3.1.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Cancel the common factor of $3 x + \frac{3}{2}$ and $3$ .

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Step 5.2.3.1

Factor $3$ out of $3 x$ .

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

Step 5.2.3.2

Factor $3$ out of $\frac{3}{2}$ .

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

Step 5.2.3.3

Factor $3$ out of $3 (x) + 3 (\frac{1}{2})$ .

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

Step 5.2.3.4

Cancel the common factors.

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Step 5.2.3.4.1

Factor $3$ out of $3$ .

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

Step 5.2.3.4.2

Cancel the common factor.

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

Step 5.2.3.4.3

Rewrite the expression.

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

Step 5.2.3.4.4

Divide $x + \frac{1}{2}$ by $1$ .

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

Step 5.2.4

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

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Step 5.2.4.1

Combine the numerators over the common denominator.

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

Step 5.2.4.2

Subtract $1$ from $1$ .

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

Step 5.2.4.3

Divide $0$ by $2$ .

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

Step 5.2.4.4

Add $x$ and $0$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

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.

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

Step 5.3.3.2.2

Rewrite the expression.

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

Step 5.3.3.3

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

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Step 5.3.3.3.1

Multiply $- 1$ by $3$ .

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

Step 5.3.3.3.2

Combine $- 3$ and $\frac{1}{2}$ .

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

Step 5.3.3.4

Move the negative in front of the fraction.

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

Step 5.3.4

Combine the opposite terms in $x - \frac{3}{2} + \frac{3}{2}$ .

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Step 5.3.4.1

Add $- \frac{3}{2}$ and $\frac{3}{2}$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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