Trigonometry Examples

$g (x) = 7 + \frac{3 x}{4}$

Step 1

Write $g (x) = 7 + \frac{3 x}{4}$ as an equation.

$y = 7 + \frac{3 x}{4}$

Step 2

Interchange the variables.

$x = 7 + \frac{3 y}{4}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $7 + \frac{3 y}{4} = x$ .

$7 + \frac{3 y}{4} = x$

Step 3.2

Subtract $7$ from both sides of the equation.

$\frac{3 y}{4} = x - 7$

Step 3.3

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

$\frac{4}{3} \cdot \frac{3 y}{4} = \frac{4}{3} (x - 7)$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

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

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Step 3.4.1.1.1

Cancel the common factor of $4$ .

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Step 3.4.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 y}{4} = \frac{4}{3} (x - 7)$

Step 3.4.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{4}{3} (x - 7)$

Step 3.4.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.1.1.2.1

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{4}{3} (x - 7)$

Step 3.4.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{4}{3} (x - 7)$

Step 3.4.1.1.2.3

Rewrite the expression.

$y = \frac{4}{3} (x - 7)$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $\frac{4}{3} (x - 7)$ .

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Step 3.4.2.1.1

Apply the distributive property.

$y = \frac{4}{3} x + \frac{4}{3} \cdot - 7$

Step 3.4.2.1.2

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

$y = \frac{4 x}{3} + \frac{4}{3} \cdot - 7$

Step 3.4.2.1.3

Multiply $\frac{4}{3} \cdot - 7$ .

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Step 3.4.2.1.3.1

Combine $\frac{4}{3}$ and $- 7$ .

$y = \frac{4 x}{3} + \frac{4 \cdot - 7}{3}$

Step 3.4.2.1.3.2

Multiply $4$ by $- 7$ .

$y = \frac{4 x}{3} + \frac{- 28}{3}$

Step 3.4.2.1.4

Move the negative in front of the fraction.

$y = \frac{4 x}{3} - \frac{28}{3}$

Step 4

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

$g^{- 1} (x) = \frac{4 x}{3} - \frac{28}{3}$

Step 5

Verify if $g^{- 1} (x) = \frac{4 x}{3} - \frac{28}{3}$ is the inverse of $g (x) = 7 + \frac{3 x}{4}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $g^{- 1} (7 + \frac{3 x}{4})$ by substituting in the value of $g$ into $g^{- 1}$ .

$g^{- 1} (7 + \frac{3 x}{4}) = \frac{4 (7 + \frac{3 x}{4})}{3} - \frac{28}{3}$

Step 5.2.3

Combine the numerators over the common denominator.

$g^{- 1} (7 + \frac{3 x}{4}) = \frac{4 (7 + \frac{3 x}{4}) - 28}{3}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Apply the distributive property.

$g^{- 1} (7 + \frac{3 x}{4}) = \frac{4 \cdot 7 + 4 (\frac{3 x}{4}) - 28}{3}$

Step 5.2.4.2

Multiply $4$ by $7$ .

$g^{- 1} (7 + \frac{3 x}{4}) = \frac{28 + 4 (\frac{3 x}{4}) - 28}{3}$

Step 5.2.4.3

Cancel the common factor of $4$ .

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Step 5.2.4.3.1

Cancel the common factor.

$g^{- 1} (7 + \frac{3 x}{4}) = \frac{28 + 4 (\frac{3 x}{4}) - 28}{3}$

Step 5.2.4.3.2

Rewrite the expression.

$g^{- 1} (7 + \frac{3 x}{4}) = \frac{28 + 3 x - 28}{3}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $28 + 3 x - 28$ .

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Step 5.2.5.1.1

Subtract $28$ from $28$ .

$g^{- 1} (7 + \frac{3 x}{4}) = \frac{3 x + 0}{3}$

Step 5.2.5.1.2

Add $3 x$ and $0$ .

$g^{- 1} (7 + \frac{3 x}{4}) = \frac{3 x}{3}$

Step 5.2.5.2

Cancel the common factor of $3$ .

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Step 5.2.5.2.1

Cancel the common factor.

$g^{- 1} (7 + \frac{3 x}{4}) = \frac{3 x}{3}$

Step 5.2.5.2.2

Divide $x$ by $1$ .

$g^{- 1} (7 + \frac{3 x}{4}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $g (\frac{4 x}{3} - \frac{28}{3})$ by substituting in the value of $g^{- 1}$ into $g$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{3 (\frac{4 x}{3} - \frac{28}{3})}{4}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the numerator.

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Step 5.3.3.1.1

Combine the numerators over the common denominator.

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{3 (\frac{4 x - 28}{3})}{4}$

Step 5.3.3.1.2

Factor $4$ out of $4 x - 28$ .

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Step 5.3.3.1.2.1

Factor $4$ out of $4 x$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{3 (\frac{4 (x) - 28}{3})}{4}$

Step 5.3.3.1.2.2

Factor $4$ out of $- 28$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{3 (\frac{4 x + 4 \cdot - 7}{3})}{4}$

Step 5.3.3.1.2.3

Factor $4$ out of $4 x + 4 \cdot - 7$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{3 (\frac{4 (x - 7)}{3})}{4}$

Step 5.3.3.2

Combine $3$ and $\frac{4 (x - 7)}{3}$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{\frac{3 (4 (x - 7))}{3}}{4}$

Step 5.3.3.3

Multiply $3$ by $4$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{\frac{12 (x - 7)}{3}}{4}$

Step 5.3.3.4

Reduce the expression by cancelling the common factors.

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Step 5.3.3.4.1

Reduce the expression $\frac{12 (x - 7)}{3}$ by cancelling the common factors.

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Step 5.3.3.4.1.1

Factor $3$ out of $12 (x - 7)$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{\frac{3 (4 (x - 7))}{3}}{4}$

Step 5.3.3.4.1.2

Factor $3$ out of $3$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{\frac{3 (4 (x - 7))}{3 (1)}}{4}$

Step 5.3.3.4.1.3

Cancel the common factor.

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{\frac{3 (4 (x - 7))}{3 \cdot 1}}{4}$

Step 5.3.3.4.1.4

Rewrite the expression.

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{\frac{4 (x - 7)}{1}}{4}$

Step 5.3.3.4.2

Divide $4 (x - 7)$ by $1$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{4 (x - 7)}{4}$

Step 5.3.3.5

Cancel the common factor of $4$ .

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Step 5.3.3.5.1

Cancel the common factor.

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + \frac{4 (x - 7)}{4}$

Step 5.3.3.5.2

Divide $x - 7$ by $1$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = 7 + x - 7$

Step 5.3.4

Combine the opposite terms in $7 + x - 7$ .

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Step 5.3.4.1

Subtract $7$ from $7$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$g (\frac{4 x}{3} - \frac{28}{3}) = x$

Step 5.4

Since $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ , then $g^{- 1} (x) = \frac{4 x}{3} - \frac{28}{3}$ is the inverse of $g (x) = 7 + \frac{3 x}{4}$ .

$g^{- 1} (x) = \frac{4 x}{3} - \frac{28}{3}$