Trigonometry Examples

$g (x) = \frac{x^{3}}{8} + 16$

Step 1

Write $g (x) = \frac{x^{3}}{8} + 16$ as an equation.

$y = \frac{x^{3}}{8} + 16$

Step 2

Interchange the variables.

$x = \frac{y^{3}}{8} + 16$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y^{3}}{8} + 16 = x$ .

$\frac{y^{3}}{8} + 16 = x$

Step 3.2

Subtract $16$ from both sides of the equation.

$\frac{y^{3}}{8} = x - 16$

Step 3.3

Multiply both sides of the equation by $8$ .

$8 \frac{y^{3}}{8} = 8 (x - 16)$

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

Cancel the common factor of $8$ .

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Step 3.4.1.1.1

Cancel the common factor.

$8 \frac{y^{3}}{8} = 8 (x - 16)$

Step 3.4.1.1.2

Rewrite the expression.

$y^{3} = 8 (x - 16)$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $8 (x - 16)$ .

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Step 3.4.2.1.1

Apply the distributive property.

$y^{3} = 8 x + 8 \cdot - 16$

Step 3.4.2.1.2

Multiply $8$ by $- 16$ .

$y^{3} = 8 x - 128$

Step 3.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{8 x - 128}$

Step 3.6

Simplify $\sqrt[3]{8 x - 128}$ .

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Step 3.6.1

Factor $8$ out of $8 x - 128$ .

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Step 3.6.1.1

Factor $8$ out of $8 x$ .

$y = \sqrt[3]{8 (x) - 128}$

Step 3.6.1.2

Factor $8$ out of $- 128$ .

$y = \sqrt[3]{8 x + 8 \cdot - 16}$

Step 3.6.1.3

Factor $8$ out of $8 x + 8 \cdot - 16$ .

$y = \sqrt[3]{8 (x - 16)}$

Step 3.6.2

Rewrite $8$ as $2^{3}$ .

$y = \sqrt[3]{2^{3} (x - 16)}$

Step 3.6.3

Pull terms out from under the radical.

$y = 2 \sqrt[3]{x - 16}$

Step 4

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

$g^{- 1} (x) = 2 \sqrt[3]{x - 16}$

Step 5

Verify if $g^{- 1} (x) = 2 \sqrt[3]{x - 16}$ is the inverse of $g (x) = \frac{x^{3}}{8} + 16$ .

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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 $g^{- 1} (\frac{x^{3}}{8} + 16)$ by substituting in the value of $g$ into $g^{- 1}$ .

$g^{- 1} (\frac{x^{3}}{8} + 16) = 2 \sqrt[3]{(\frac{x^{3}}{8} + 16) - 16}$

Step 5.2.3

Subtract $16$ from $16$ .

$g^{- 1} (\frac{x^{3}}{8} + 16) = 2 \sqrt[3]{\frac{x^{3}}{8} + 0}$

Step 5.2.4

Add $\frac{x^{3}}{8}$ and $0$ .

$g^{- 1} (\frac{x^{3}}{8} + 16) = 2 \sqrt[3]{\frac{x^{3}}{8}}$

Step 5.2.5

Rewrite $8$ as $2^{3}$ .

$g^{- 1} (\frac{x^{3}}{8} + 16) = 2 \sqrt[3]{\frac{x^{3}}{2^{3}}}$

Step 5.2.6

Rewrite $\frac{x^{3}}{2^{3}}$ as ${(\frac{x}{2})}^{3}$ .

$g^{- 1} (\frac{x^{3}}{8} + 16) = 2 \sqrt[3]{{(\frac{x}{2})}^{3}}$

Step 5.2.7

Pull terms out from under the radical, assuming real numbers.

$g^{- 1} (\frac{x^{3}}{8} + 16) = 2 (\frac{x}{2})$

Step 5.2.8

Cancel the common factor of $2$ .

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Step 5.2.8.1

Cancel the common factor.

$g^{- 1} (\frac{x^{3}}{8} + 16) = 2 (\frac{x}{2})$

Step 5.2.8.2

Rewrite the expression.

$g^{- 1} (\frac{x^{3}}{8} + 16) = x$

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 $g (2 \sqrt[3]{x - 16})$ by substituting in the value of $g^{- 1}$ into $g$ .

$g (2 \sqrt[3]{x - 16}) = \frac{{(2 \sqrt[3]{x - 16})}^{3}}{8} + 16$

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

Apply the product rule to $2 \sqrt[3]{x - 16}$ .

$g (2 \sqrt[3]{x - 16}) = \frac{2^{3} {\sqrt[3]{x - 16}}^{3}}{8} + 16$

Step 5.3.3.1.2

Raise $2$ to the power of $3$ .

$g (2 \sqrt[3]{x - 16}) = \frac{8 {\sqrt[3]{x - 16}}^{3}}{8} + 16$

Step 5.3.3.1.3

Rewrite ${\sqrt[3]{x - 16}}^{3}$ as $x - 16$ .

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Step 5.3.3.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 16}$ as ${(x - 16)}^{\frac{1}{3}}$ .

$g (2 \sqrt[3]{x - 16}) = \frac{8 {({(x - 16)}^{\frac{1}{3}})}^{3}}{8} + 16$

Step 5.3.3.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$g (2 \sqrt[3]{x - 16}) = \frac{8 {(x - 16)}^{\frac{1}{3} \cdot 3}}{8} + 16$

Step 5.3.3.1.3.3

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

$g (2 \sqrt[3]{x - 16}) = \frac{8 {(x - 16)}^{\frac{3}{3}}}{8} + 16$

Step 5.3.3.1.3.4

Cancel the common factor of $3$ .

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Step 5.3.3.1.3.4.1

Cancel the common factor.

$g (2 \sqrt[3]{x - 16}) = \frac{8 {(x - 16)}^{\frac{3}{3}}}{8} + 16$

Step 5.3.3.1.3.4.2

Rewrite the expression.

$g (2 \sqrt[3]{x - 16}) = \frac{8 (x - 16)}{8} + 16$

Step 5.3.3.1.3.5

Simplify.

$g (2 \sqrt[3]{x - 16}) = \frac{8 (x - 16)}{8} + 16$

Step 5.3.3.2

Cancel the common factor of $8$ .

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Step 5.3.3.2.1

Cancel the common factor.

$g (2 \sqrt[3]{x - 16}) = \frac{8 (x - 16)}{8} + 16$

Step 5.3.3.2.2

Divide $x - 16$ by $1$ .

$g (2 \sqrt[3]{x - 16}) = x - 16 + 16$

Step 5.3.4

Combine the opposite terms in $x - 16 + 16$ .

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Step 5.3.4.1

Add $- 16$ and $16$ .

$g (2 \sqrt[3]{x - 16}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$g (2 \sqrt[3]{x - 16}) = x$

Step 5.4

Since $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ , then $g^{- 1} (x) = 2 \sqrt[3]{x - 16}$ is the inverse of $g (x) = \frac{x^{3}}{8} + 16$ .

$g^{- 1} (x) = 2 \sqrt[3]{x - 16}$