Trigonometry Examples

$\frac{3}{4} x + \frac{3}{4}$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Subtract $\frac{3}{4}$ from both sides of the equation.

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

Step 2.4

Multiply both sides of the equation by $\frac{4}{3}$ .

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

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 $\frac{4}{3} \cdot \frac{3 y}{4}$ .

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Step 2.5.1.1.1

Cancel the common factor of $4$ .

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Step 2.5.1.1.1.1

Cancel the common factor.

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

Step 2.5.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{4}{3} (x - \frac{3}{4})$

Step 2.5.1.1.2

Cancel the common factor of $3$ .

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Step 2.5.1.1.2.1

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{4}{3} (x - \frac{3}{4})$

Step 2.5.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{4}{3} (x - \frac{3}{4})$

Step 2.5.1.1.2.3

Rewrite the expression.

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

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify $\frac{4}{3} (x - \frac{3}{4})$ .

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Step 2.5.2.1.1

Apply the distributive property.

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

Step 2.5.2.1.2

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

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

Step 2.5.2.1.3

Cancel the common factor of $4$ .

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Step 2.5.2.1.3.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

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

Step 2.5.2.1.3.2

Cancel the common factor.

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

Step 2.5.2.1.3.3

Rewrite the expression.

$y = \frac{4 x}{3} + \frac{1}{3} \cdot - 3$

Step 2.5.2.1.4

Cancel the common factor of $3$ .

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Step 2.5.2.1.4.1

Factor $3$ out of $- 3$ .

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

Step 2.5.2.1.4.2

Cancel the common factor.

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

Step 2.5.2.1.4.3

Rewrite the expression.

$y = \frac{4 x}{3} - 1$

Step 3

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

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

Step 4

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

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

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

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

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

Step 4.2.3.1.2

Combine the numerators over the common denominator.

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

Step 4.2.3.1.3

Factor $3$ out of $3 x + 3$ .

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Step 4.2.3.1.3.1

Factor $3$ out of $3 x$ .

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

Step 4.2.3.1.3.2

Factor $3$ out of $3$ .

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

Step 4.2.3.1.3.3

Factor $3$ out of $3 (x) + 3 (1)$ .

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

Step 4.2.3.2

Combine $4$ and $\frac{3 (x + 1)}{4}$ .

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

Step 4.2.3.3

Multiply $4$ by $3$ .

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

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 $\frac{12 (x + 1)}{4}$ by cancelling the common factors.

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Step 4.2.3.4.1.1

Factor $4$ out of $12 (x + 1)$ .

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

Step 4.2.3.4.1.2

Factor $4$ out of $4$ .

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

Step 4.2.3.4.1.3

Cancel the common factor.

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

Step 4.2.3.4.1.4

Rewrite the expression.

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

Step 4.2.3.4.2

Divide $3 (x + 1)$ by $1$ .

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

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.

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

Step 4.2.3.5.2

Divide $x + 1$ by $1$ .

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

Step 4.2.4

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

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Step 4.2.4.1

Subtract $1$ from $1$ .

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

Step 4.2.4.2

Add $x$ and $0$ .

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

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

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

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

Step 4.3.3.2

Cancel the common factor of $3$ .

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Step 4.3.3.2.1

Cancel the common factor.

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

Step 4.3.3.2.2

Rewrite the expression.

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

Step 4.3.3.3

Cancel the common factor of $4$ .

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Step 4.3.3.3.1

Factor $4$ out of $4 x$ .

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

Step 4.3.3.3.2

Cancel the common factor.

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

Step 4.3.3.3.3

Rewrite the expression.

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

Step 4.3.3.4

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

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Step 4.3.3.4.1

Combine $\frac{3}{4}$ and $- 1$ .

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

Step 4.3.3.4.2

Multiply $3$ by $- 1$ .

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

Step 4.3.3.5

Move the negative in front of the fraction.

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

Step 4.3.4

Combine the opposite terms in $x - \frac{3}{4} + \frac{3}{4}$ .

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Step 4.3.4.1

Add $- \frac{3}{4}$ and $\frac{3}{4}$ .

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

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

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

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