Trigonometry Examples

$y = - 3 x^{3}$

Step 1

Interchange the variables.

$x = - 3 y^{3}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- 3 y^{3} = x$ .

$- 3 y^{3} = x$

Step 2.2

Divide each term in $- 3 y^{3} = x$ by $- 3$ and simplify.

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Step 2.2.1

Divide each term in $- 3 y^{3} = x$ by $- 3$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $y^{3}$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

Rewrite $- \frac{x}{3}$ as ${({(- 1)}^{3})}^{3} \frac{x}{3}$ .

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Step 2.4.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 2.4.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

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

Step 2.4.2

Pull terms out from under the radical.

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

Step 2.4.3

Raise $- 1$ to the power of $3$ .

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

Step 2.4.4

Rewrite $\sqrt[3]{\frac{x}{3}}$ as $\frac{\sqrt[3]{x}}{\sqrt[3]{3}}$ .

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

Step 2.4.5

Multiply $\frac{\sqrt[3]{x}}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

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

Step 2.4.6

Combine and simplify the denominator.

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Step 2.4.6.1

Multiply $\frac{\sqrt[3]{x}}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

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

Step 2.4.6.2

Raise $\sqrt[3]{3}$ to the power of $1$ .

$y = - \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1} {\sqrt[3]{3}}^{2}}$

Step 2.4.6.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = - \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1 + 2}}$

Step 2.4.6.4

Add $1$ and $2$ .

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

Step 2.4.6.5

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

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Step 2.4.6.5.1

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

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

Step 2.4.6.5.2

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

$y = - \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{3^{\frac{1}{3} \cdot 3}}$

Step 2.4.6.5.3

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

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

Step 2.4.6.5.4

Cancel the common factor of $3$ .

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Step 2.4.6.5.4.1

Cancel the common factor.

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

Step 2.4.6.5.4.2

Rewrite the expression.

$y = - \frac{\sqrt[3]{x} {\sqrt[3]{3}}^{2}}{3^{1}}$

Step 2.4.6.5.5

Evaluate the exponent.

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

Step 2.4.7

Simplify the numerator.

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Step 2.4.7.1

Rewrite ${\sqrt[3]{3}}^{2}$ as $\sqrt[3]{3^{2}}$ .

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

Step 2.4.7.2

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

$y = - \frac{\sqrt[3]{x} \sqrt[3]{9}}{3}$

Step 2.4.8

Simplify with factoring out.

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Step 2.4.8.1

Combine using the product rule for radicals.

$y = - \frac{\sqrt[3]{x \cdot 9}}{3}$

Step 2.4.8.2

Reorder factors in $- \frac{\sqrt[3]{x \cdot 9}}{3}$ .

$y = - \frac{\sqrt[3]{9 x}}{3}$

Step 3

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

$f^{- 1} (x) = - \frac{\sqrt[3]{9 x}}{3}$

Step 4

Verify if $f^{- 1} (x) = - \frac{\sqrt[3]{9 x}}{3}$ is the inverse of $f (x) = - 3 x^{3}$ .

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

$f^{- 1} (- 3 x^{3}) = - \frac{\sqrt[3]{9 (- 3 x^{3})}}{3}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Multiply $- 3$ by $9$ .

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

Step 4.2.3.2

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

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

Step 4.2.3.3

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

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

Step 4.2.4

Cancel the common factor of $- 3$ and $3$ .

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Step 4.2.4.1

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

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

Step 4.2.4.2

Cancel the common factors.

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Step 4.2.4.2.1

Factor $3$ out of $3$ .

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

Step 4.2.4.2.2

Cancel the common factor.

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

Step 4.2.4.2.3

Rewrite the expression.

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

Step 4.2.4.2.4

Divide $- x$ by $1$ .

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

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 $f (- \frac{\sqrt[3]{9 x}}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 {(- \frac{\sqrt[3]{9 x}}{3})}^{3}$

Step 4.3.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.3.3.1

Apply the product rule to $- \frac{\sqrt[3]{9 x}}{3}$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 ({(- 1)}^{3} {(\frac{\sqrt[3]{9 x}}{3})}^{3})$

Step 4.3.3.2

Apply the product rule to $\frac{\sqrt[3]{9 x}}{3}$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 ({(- 1)}^{3} (\frac{{\sqrt[3]{9 x}}^{3}}{3^{3}}))$

Step 4.3.4

Raise $- 1$ to the power of $3$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 (- \frac{{\sqrt[3]{9 x}}^{3}}{3^{3}})$

Step 4.3.5

Rewrite ${\sqrt[3]{9 x}}^{3}$ as $9 x$ .

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Step 4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 x}$ as ${(9 x)}^{\frac{1}{3}}$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 (- \frac{{({(9 x)}^{\frac{1}{3}})}^{3}}{3^{3}})$

Step 4.3.5.2

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

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 (- \frac{{(9 x)}^{\frac{1}{3} \cdot 3}}{3^{3}})$

Step 4.3.5.3

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

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 (- \frac{{(9 x)}^{\frac{3}{3}}}{3^{3}})$

Step 4.3.5.4

Cancel the common factor of $3$ .

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Step 4.3.5.4.1

Cancel the common factor.

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 (- \frac{{(9 x)}^{\frac{3}{3}}}{3^{3}})$

Step 4.3.5.4.2

Rewrite the expression.

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 (- \frac{9 x}{3^{3}})$

Step 4.3.5.5

Simplify.

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 (- \frac{9 x}{3^{3}})$

Step 4.3.6

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

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 (- \frac{9 x}{27})$

Step 4.3.7

Cancel the common factor of $3$ .

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Step 4.3.7.1

Move the leading negative in $- \frac{9 x}{27}$ into the numerator.

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 3 \frac{- 9 x}{27}$

Step 4.3.7.2

Factor $3$ out of $- 3$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = 3 (- 1) (\frac{- 9 x}{27})$

Step 4.3.7.3

Factor $3$ out of $27$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = 3 \cdot (- 1 \frac{- 9 x}{3 \cdot 9})$

Step 4.3.7.4

Cancel the common factor.

$f (- \frac{\sqrt[3]{9 x}}{3}) = 3 \cdot (- 1 \frac{- 9 x}{3 \cdot 9})$

Step 4.3.7.5

Rewrite the expression.

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 1 \frac{- 9 x}{9}$

Step 4.3.8

Cancel the common factor of $- 9$ and $9$ .

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Step 4.3.8.1

Factor $9$ out of $- 9 x$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 1 \frac{9 (- x)}{9}$

Step 4.3.8.2

Cancel the common factors.

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Step 4.3.8.2.1

Factor $9$ out of $9$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 1 \frac{9 (- x)}{9 (1)}$

Step 4.3.8.2.2

Cancel the common factor.

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 1 \frac{9 (- x)}{9 \cdot 1}$

Step 4.3.8.2.3

Rewrite the expression.

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 1 \frac{- x}{1}$

Step 4.3.8.2.4

Divide $- x$ by $1$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = - 1 (- x)$

Step 4.3.9

Multiply $- 1 (- x)$ .

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Step 4.3.9.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = 1 x$

Step 4.3.9.2

Multiply $x$ by $1$ .

$f (- \frac{\sqrt[3]{9 x}}{3}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{\sqrt[3]{9 x}}{3}$ is the inverse of $f (x) = - 3 x^{3}$ .

$f^{- 1} (x) = - \frac{\sqrt[3]{9 x}}{3}$