Trigonometry Examples

$f (x) = \frac{\sqrt[3]{x - 7}}{3}$

Step 1

Write $f (x) = \frac{\sqrt[3]{x - 7}}{3}$ as an equation.

$y = \frac{\sqrt[3]{x - 7}}{3}$

Step 2

Interchange the variables.

$x = \frac{\sqrt[3]{y - 7}}{3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{\sqrt[3]{y - 7}}{3} = x$ .

$\frac{\sqrt[3]{y - 7}}{3} = x$

Step 3.2

Multiply both sides of the equation by $3$ .

$3 \frac{\sqrt[3]{y - 7}}{3} = 3 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $3$ .

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Step 3.3.1.1

Cancel the common factor.

$3 \frac{\sqrt[3]{y - 7}}{3} = 3 x$

Step 3.3.1.2

Rewrite the expression.

$\sqrt[3]{y - 7} = 3 x$

Step 3.4

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{y - 7}}^{3} = {(3 x)}^{3}$

Step 3.5

Simplify each side of the equation.

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y - 7}$ as ${(y - 7)}^{\frac{1}{3}}$ .

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

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Simplify ${({(y - 7)}^{\frac{1}{3}})}^{3}$ .

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Step 3.5.2.1.1

Multiply the exponents in ${({(y - 7)}^{\frac{1}{3}})}^{3}$ .

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Step 3.5.2.1.1.1

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

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

Step 3.5.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.5.2.1.1.2.1

Cancel the common factor.

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

Step 3.5.2.1.1.2.2

Rewrite the expression.

${(y - 7)}^{1} = {(3 x)}^{3}$

Step 3.5.2.1.2

Simplify.

$y - 7 = {(3 x)}^{3}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify ${(3 x)}^{3}$ .

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Step 3.5.3.1.1

Apply the product rule to $3 x$ .

$y - 7 = 3^{3} x^{3}$

Step 3.5.3.1.2

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

$y - 7 = 27 x^{3}$

Step 3.6

Add $7$ to both sides of the equation.

$y = 27 x^{3} + 7$

Step 4

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

$f^{- 1} (x) = 27 x^{3} + 7$

Step 5

Verify if $f^{- 1} (x) = 27 x^{3} + 7$ is the inverse of $f (x) = \frac{\sqrt[3]{x - 7}}{3}$ .

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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 $f^{- 1} (\frac{\sqrt[3]{x - 7}}{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = 27 {(\frac{\sqrt[3]{x - 7}}{3})}^{3} + 7$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Apply the product rule to $\frac{\sqrt[3]{x - 7}}{3}$ .

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = 27 (\frac{{\sqrt[3]{x - 7}}^{3}}{3^{3}}) + 7$

Step 5.2.3.2

Rewrite ${\sqrt[3]{x - 7}}^{3}$ as $x - 7$ .

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Step 5.2.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 7}$ as ${(x - 7)}^{\frac{1}{3}}$ .

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = 27 (\frac{{({(x - 7)}^{\frac{1}{3}})}^{3}}{3^{3}}) + 7$

Step 5.2.3.2.2

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

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = 27 (\frac{{(x - 7)}^{\frac{1}{3} \cdot 3}}{3^{3}}) + 7$

Step 5.2.3.2.3

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

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = 27 (\frac{{(x - 7)}^{\frac{3}{3}}}{3^{3}}) + 7$

Step 5.2.3.2.4

Cancel the common factor of $3$ .

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Step 5.2.3.2.4.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = 27 (\frac{{(x - 7)}^{\frac{3}{3}}}{3^{3}}) + 7$

Step 5.2.3.2.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = 27 (\frac{x - 7}{3^{3}}) + 7$

Step 5.2.3.2.5

Simplify.

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = 27 (\frac{x - 7}{3^{3}}) + 7$

Step 5.2.3.3

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

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = 27 (\frac{x - 7}{27}) + 7$

Step 5.2.3.4

Cancel the common factor of $27$ .

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Step 5.2.3.4.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = 27 (\frac{x - 7}{27}) + 7$

Step 5.2.3.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = x - 7 + 7$

Step 5.2.4

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

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Step 5.2.4.1

Add $- 7$ and $7$ .

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[3]{x - 7}}{3}) = x$

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 $f (27 x^{3} + 7)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (27 x^{3} + 7) = \frac{\sqrt[3]{(27 x^{3} + 7) - 7}}{3}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Subtract $7$ from $7$ .

$f (27 x^{3} + 7) = \frac{\sqrt[3]{27 x^{3} + 0}}{3}$

Step 5.3.3.2

Add $27 x^{3}$ and $0$ .

$f (27 x^{3} + 7) = \frac{\sqrt[3]{27 x^{3}}}{3}$

Step 5.3.3.3

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

$f (27 x^{3} + 7) = \frac{\sqrt[3]{{(3 x)}^{3}}}{3}$

Step 5.3.3.4

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

$f (27 x^{3} + 7) = \frac{3 x}{3}$

Step 5.3.4

Cancel the common factor of $3$ .

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Step 5.3.4.1

Cancel the common factor.

$f (27 x^{3} + 7) = \frac{3 x}{3}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (27 x^{3} + 7) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 27 x^{3} + 7$ is the inverse of $f (x) = \frac{\sqrt[3]{x - 7}}{3}$ .

$f^{- 1} (x) = 27 x^{3} + 7$