Trigonometry Examples

$\frac{3 x - 6}{4}$

Step 1

Interchange the variables.

$x = \frac{3 y - 6}{4}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{3 y - 6}{4} = x$ .

$\frac{3 y - 6}{4} = x$

Step 2.2

Multiply both sides by $4$ .

$\frac{3 y - 6}{4} \cdot 4 = x \cdot 4$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{3 y - 6}{4} \cdot 4$ .

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Step 2.3.1.1.1

Factor $3$ out of $3 y - 6$ .

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Step 2.3.1.1.1.1

Factor $3$ out of $3 y$ .

$\frac{3 (y) - 6}{4} \cdot 4 = x \cdot 4$

Step 2.3.1.1.1.2

Factor $3$ out of $- 6$ .

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

Step 2.3.1.1.1.3

Factor $3$ out of $3 y + 3 \cdot - 2$ .

$\frac{3 (y - 2)}{4} \cdot 4 = x \cdot 4$

Step 2.3.1.1.2

Cancel the common factor of $4$ .

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Step 2.3.1.1.2.1

Cancel the common factor.

$\frac{3 (y - 2)}{4} \cdot 4 = x \cdot 4$

Step 2.3.1.1.2.2

Rewrite the expression.

$3 (y - 2) = x \cdot 4$

Step 2.3.1.1.3

Apply the distributive property.

$3 y + 3 \cdot - 2 = x \cdot 4$

Step 2.3.1.1.4

Multiply $3$ by $- 2$ .

$3 y - 6 = x \cdot 4$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Move $4$ to the left of $x$ .

$3 y - 6 = 4 x$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Add $6$ to both sides of the equation.

$3 y = 4 x + 6$

Step 2.4.2

Divide each term in $3 y = 4 x + 6$ by $3$ and simplify.

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Step 2.4.2.1

Divide each term in $3 y = 4 x + 6$ by $3$ .

$\frac{3 y}{3} = \frac{4 x}{3} + \frac{6}{3}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{4 x}{3} + \frac{6}{3}$

Step 2.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4 x}{3} + \frac{6}{3}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $6$ by $3$ .

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

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = \frac{4 x}{3} + 2$ is the inverse of $f (x) = \frac{3 x - 6}{4}$ .

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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 $f^{- 1} (\frac{3 x - 6}{4})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Factor $3$ out of $3 x - 6$ .

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Step 4.2.3.1.1

Factor $3$ out of $3 x$ .

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

Step 4.2.3.1.2

Factor $3$ out of $- 6$ .

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

Step 4.2.3.1.3

Factor $3$ out of $3 x + 3 \cdot - 2$ .

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

Step 4.2.3.2

Combine $4$ and $\frac{3 (x - 2)}{4}$ .

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

Step 4.2.3.3

Multiply $4$ by $3$ .

$f^{- 1} (\frac{3 x - 6}{4}) = \frac{\frac{12 (x - 2)}{4}}{3} + 2$

Step 4.2.3.4

Reduce the expression by cancelling the common factors.

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Step 4.2.3.4.1

Reduce the expression $\frac{12 (x - 2)}{4}$ by cancelling the common factors.

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Step 4.2.3.4.1.1

Factor $4$ out of $12 (x - 2)$ .

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

Step 4.2.3.4.1.2

Factor $4$ out of $4$ .

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

Step 4.2.3.4.1.3

Cancel the common factor.

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

Step 4.2.3.4.1.4

Rewrite the expression.

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

Step 4.2.3.4.2

Divide $3 (x - 2)$ by $1$ .

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

Step 4.2.3.5

Cancel the common factor of $3$ .

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Step 4.2.3.5.1

Cancel the common factor.

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

Step 4.2.3.5.2

Divide $x - 2$ by $1$ .

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

Step 4.2.4

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

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Step 4.2.4.1

Add $- 2$ and $2$ .

$f^{- 1} (\frac{3 x - 6}{4}) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

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

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

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

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

Factor $3$ out of $3 (\frac{4 x}{3} + 2) - 6$ .

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Step 4.3.3.1.1

Factor $3$ out of $- 6$ .

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

Step 4.3.3.1.2

Factor $3$ out of $3 (\frac{4 x}{3} + 2) + 3 \cdot - 2$ .

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

Step 4.3.3.2

Subtract $2$ from $2$ .

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

Step 4.3.3.3

Add $\frac{4 x}{3}$ and $0$ .

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

Step 4.3.4

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

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

Step 4.3.5

Multiply $3$ by $4$ .

$f (\frac{4 x}{3} + 2) = \frac{\frac{12 x}{3}}{4}$

Step 4.3.6

Reduce the expression by cancelling the common factors.

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Step 4.3.6.1

Reduce the expression $\frac{12 x}{3}$ by cancelling the common factors.

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Step 4.3.6.1.1

Factor $3$ out of $12 x$ .

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

Step 4.3.6.1.2

Factor $3$ out of $3$ .

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

Step 4.3.6.1.3

Cancel the common factor.

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

Step 4.3.6.1.4

Rewrite the expression.

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

Step 4.3.6.2

Divide $4 x$ by $1$ .

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

Step 4.3.7

Cancel the common factor of $4$ .

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Step 4.3.7.1

Cancel the common factor.

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

Step 4.3.7.2

Divide $x$ by $1$ .

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

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{4 x}{3} + 2$ is the inverse of $f (x) = \frac{3 x - 6}{4}$ .

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