Trigonometry Examples

$f (x) = 2 - 7 x^{3}$

Step 1

Write $f (x) = 2 - 7 x^{3}$ as an equation.

$y = 2 - 7 x^{3}$

Step 2

Interchange the variables.

$x = 2 - 7 y^{3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 - 7 y^{3} = x$ .

$2 - 7 y^{3} = x$

Step 3.2

Subtract $2$ from both sides of the equation.

$- 7 y^{3} = x - 2$

Step 3.3

Divide each term in $- 7 y^{3} = x - 2$ by $- 7$ and simplify.

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Step 3.3.1

Divide each term in $- 7 y^{3} = x - 2$ by $- 7$ .

$\frac{- 7 y^{3}}{- 7} = \frac{x}{- 7} + \frac{- 2}{- 7}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 7$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 7 y^{3}}{- 7} = \frac{x}{- 7} + \frac{- 2}{- 7}$

Step 3.3.2.1.2

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

$y^{3} = \frac{x}{- 7} + \frac{- 2}{- 7}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move the negative in front of the fraction.

$y^{3} = - \frac{x}{7} + \frac{- 2}{- 7}$

Step 3.3.3.1.2

Dividing two negative values results in a positive value.

$y^{3} = - \frac{x}{7} + \frac{2}{7}$

Step 3.4

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

$y = \sqrt[3]{- \frac{x}{7} + \frac{2}{7}}$

Step 3.5

Simplify $\sqrt[3]{- \frac{x}{7} + \frac{2}{7}}$ .

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Step 3.5.1

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{- x + 2}{7}}$

Step 3.5.2

Rewrite $\sqrt[3]{\frac{- x + 2}{7}}$ as $\frac{\sqrt[3]{- x + 2}}{\sqrt[3]{7}}$ .

$y = \frac{\sqrt[3]{- x + 2}}{\sqrt[3]{7}}$

Step 3.5.3

Multiply $\frac{\sqrt[3]{- x + 2}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$y = \frac{\sqrt[3]{- x + 2}}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$

Step 3.5.4

Combine and simplify the denominator.

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Step 3.5.4.1

Multiply $\frac{\sqrt[3]{- x + 2}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$y = \frac{\sqrt[3]{- x + 2} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}}$

Step 3.5.4.2

Raise $\sqrt[3]{7}$ to the power of $1$ .

$y = \frac{\sqrt[3]{- x + 2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1} {\sqrt[3]{7}}^{2}}$

Step 3.5.4.3

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

$y = \frac{\sqrt[3]{- x + 2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}}$

Step 3.5.4.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{- x + 2} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}}$

Step 3.5.4.5

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

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Step 3.5.4.5.1

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

$y = \frac{\sqrt[3]{- x + 2} {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}}$

Step 3.5.4.5.2

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

$y = \frac{\sqrt[3]{- x + 2} {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}}$

Step 3.5.4.5.3

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

$y = \frac{\sqrt[3]{- x + 2} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 3.5.4.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{- x + 2} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 3.5.4.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{- x + 2} {\sqrt[3]{7}}^{2}}{7^{1}}$

Step 3.5.4.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{- x + 2} {\sqrt[3]{7}}^{2}}{7}$

Step 3.5.5

Simplify the numerator.

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Step 3.5.5.1

Rewrite ${\sqrt[3]{7}}^{2}$ as $\sqrt[3]{7^{2}}$ .

$y = \frac{\sqrt[3]{- x + 2} \sqrt[3]{7^{2}}}{7}$

Step 3.5.5.2

Raise $7$ to the power of $2$ .

$y = \frac{\sqrt[3]{- x + 2} \sqrt[3]{49}}{7}$

Step 3.5.6

Simplify with factoring out.

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Step 3.5.6.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(- x + 2) \cdot 49}}{7}$

Step 3.5.6.2

Reorder factors in $\frac{\sqrt[3]{(- x + 2) \cdot 49}}{7}$ .

$y = \frac{\sqrt[3]{49 (- x + 2)}}{7}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[3]{49 (- x + 2)}}{7}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[3]{49 (- x + 2)}}{7}$ is the inverse of $f (x) = 2 - 7 x^{3}$ .

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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 $f^{- 1} (2 - 7 x^{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 - 7 x^{3}) = \frac{\sqrt[3]{49 (- (2 - 7 x^{3}) + 2)}}{7}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Apply the distributive property.

$f^{- 1} (2 - 7 x^{3}) = \frac{\sqrt[3]{49 (- 1 \cdot 2 - (- 7 x^{3}) + 2)}}{7}$

Step 5.2.3.2

Multiply $- 1$ by $2$ .

$f^{- 1} (2 - 7 x^{3}) = \frac{\sqrt[3]{49 (- 2 - (- 7 x^{3}) + 2)}}{7}$

Step 5.2.3.3

Multiply $- 7$ by $- 1$ .

$f^{- 1} (2 - 7 x^{3}) = \frac{\sqrt[3]{49 (- 2 + 7 x^{3} + 2)}}{7}$

Step 5.2.3.4

Add $- 2$ and $2$ .

$f^{- 1} (2 - 7 x^{3}) = \frac{\sqrt[3]{49 (7 x^{3} + 0)}}{7}$

Step 5.2.3.5

Add $7 x^{3}$ and $0$ .

$f^{- 1} (2 - 7 x^{3}) = \frac{\sqrt[3]{49 \cdot (7 x^{3})}}{7}$

Step 5.2.3.6

Multiply $49$ by $7$ .

$f^{- 1} (2 - 7 x^{3}) = \frac{\sqrt[3]{343 x^{3}}}{7}$

Step 5.2.3.7

Rewrite $343 x^{3}$ as ${(7 x)}^{3}$ .

$f^{- 1} (2 - 7 x^{3}) = \frac{\sqrt[3]{{(7 x)}^{3}}}{7}$

Step 5.2.3.8

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

$f^{- 1} (2 - 7 x^{3}) = \frac{7 x}{7}$

Step 5.2.4

Cancel the common factor of $7$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (2 - 7 x^{3}) = \frac{7 x}{7}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (2 - 7 x^{3}) = x$

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 $f (\frac{\sqrt[3]{49 (- x + 2)}}{7})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 {(\frac{\sqrt[3]{49 (- x + 2)}}{7})}^{3}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the product rule to $\frac{\sqrt[3]{49 (- x + 2)}}{7}$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{{\sqrt[3]{49 (- x + 2)}}^{3}}{7^{3}}$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

Rewrite ${\sqrt[3]{49 (- x + 2)}}^{3}$ as $49 (- x + 2)$ .

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Step 5.3.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{49 (- x + 2)}$ as ${(49 (- x + 2))}^{\frac{1}{3}}$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{{({(49 (- x + 2))}^{\frac{1}{3}})}^{3}}{7^{3}}$

Step 5.3.3.2.1.2

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

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{{(49 (- x + 2))}^{\frac{1}{3} \cdot 3}}{7^{3}}$

Step 5.3.3.2.1.3

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

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{{(49 (- x + 2))}^{\frac{3}{3}}}{7^{3}}$

Step 5.3.3.2.1.4

Cancel the common factor of $3$ .

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Step 5.3.3.2.1.4.1

Cancel the common factor.

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{{(49 (- x + 2))}^{\frac{3}{3}}}{7^{3}}$

Step 5.3.3.2.1.4.2

Rewrite the expression.

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{49 (- x + 2)}{7^{3}}$

Step 5.3.3.2.1.5

Simplify.

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{49 (- x + 2)}{7^{3}}$

Step 5.3.3.2.2

Apply the distributive property.

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{49 (- x) + 49 \cdot 2}{7^{3}}$

Step 5.3.3.2.3

Multiply $- 1$ by $49$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{- 49 x + 49 \cdot 2}{7^{3}}$

Step 5.3.3.2.4

Multiply $49$ by $2$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{- 49 x + 98}{7^{3}}$

Step 5.3.3.2.5

Factor $49$ out of $- 49 x + 98$ .

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Step 5.3.3.2.5.1

Factor $49$ out of $- 49 x$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{49 (- x) + 98}{7^{3}}$

Step 5.3.3.2.5.2

Factor $49$ out of $98$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{49 (- x) + 49 (2)}{7^{3}}$

Step 5.3.3.2.5.3

Factor $49$ out of $49 (- x) + 49 (2)$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{49 (- x + 2)}{7^{3}}$

Step 5.3.3.3

Raise $7$ to the power of $3$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 7 \frac{49 (- x + 2)}{343}$

Step 5.3.3.4

Cancel the common factor of $7$ .

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Step 5.3.3.4.1

Factor $7$ out of $- 7$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 + 7 (- 1) (\frac{49 (- x + 2)}{343})$

Step 5.3.3.4.2

Factor $7$ out of $343$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 + 7 \cdot (- 1 \frac{49 (- x + 2)}{7 \cdot 49})$

Step 5.3.3.4.3

Cancel the common factor.

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 + 7 \cdot (- 1 \frac{49 (- x + 2)}{7 \cdot 49})$

Step 5.3.3.4.4

Rewrite the expression.

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 1 \frac{49 (- x + 2)}{49}$

Step 5.3.3.5

Cancel the common factor of $49$ .

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Step 5.3.3.5.1

Cancel the common factor.

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 1 \frac{49 (- x + 2)}{49}$

Step 5.3.3.5.2

Divide $- x + 2$ by $1$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 1 (- x + 2)$

Step 5.3.3.6

Apply the distributive property.

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 - 1 (- x) - 1 \cdot 2$

Step 5.3.3.7

Multiply $- 1 (- x)$ .

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Step 5.3.3.7.1

Multiply $- 1$ by $- 1$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 + 1 x - 1 \cdot 2$

Step 5.3.3.7.2

Multiply $x$ by $1$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 + x - 1 \cdot 2$

Step 5.3.3.8

Multiply $- 1$ by $2$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = 2 + x - 2$

Step 5.3.4

Combine the opposite terms in $2 + x - 2$ .

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Step 5.3.4.1

Subtract $2$ from $2$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{\sqrt[3]{49 (- x + 2)}}{7}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[3]{49 (- x + 2)}}{7}$ is the inverse of $f (x) = 2 - 7 x^{3}$ .

$f^{- 1} (x) = \frac{\sqrt[3]{49 (- x + 2)}}{7}$