Calculus Examples

$f (x) = 5 x^{3} + 6$

Step 1

Write $f (x) = 5 x^{3} + 6$ as an equation.

$y = 5 x^{3} + 6$

Step 2

Interchange the variables.

$x = 5 y^{3} + 6$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $5 y^{3} + 6 = x$ .

$5 y^{3} + 6 = x$

Step 3.2

Subtract $6$ from both sides of the equation.

$5 y^{3} = x - 6$

Step 3.3

Divide each term in $5 y^{3} = x - 6$ by $5$ and simplify.

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Step 3.3.1

Divide each term in $5 y^{3} = x - 6$ by $5$ .

$\frac{5 y^{3}}{5} = \frac{x}{5} + \frac{- 6}{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{5 y^{3}}{5} = \frac{x}{5} + \frac{- 6}{5}$

Step 3.3.2.1.2

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

$y^{3} = \frac{x}{5} + \frac{- 6}{5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$y^{3} = \frac{x}{5} - \frac{6}{5}$

Step 3.4

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

$y = \sqrt[3]{\frac{x}{5} - \frac{6}{5}}$

Step 3.5

Simplify $\sqrt[3]{\frac{x}{5} - \frac{6}{5}}$ .

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Step 3.5.1

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{x - 6}{5}}$

Step 3.5.2

Rewrite $\sqrt[3]{\frac{x - 6}{5}}$ as $\frac{\sqrt[3]{x - 6}}{\sqrt[3]{5}}$ .

$y = \frac{\sqrt[3]{x - 6}}{\sqrt[3]{5}}$

Step 3.5.3

Multiply $\frac{\sqrt[3]{x - 6}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$y = \frac{\sqrt[3]{x - 6}}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$

Step 3.5.4

Combine and simplify the denominator.

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Step 3.5.4.1

Multiply $\frac{\sqrt[3]{x - 6}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$y = \frac{\sqrt[3]{x - 6} {\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Step 3.5.4.2

Raise $\sqrt[3]{5}$ to the power of $1$ .

$y = \frac{\sqrt[3]{x - 6} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}$

Step 3.5.4.3

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

$y = \frac{\sqrt[3]{x - 6} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}$

Step 3.5.4.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{x - 6} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}}$

Step 3.5.4.5

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

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Step 3.5.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{x - 6} {\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}$

Step 3.5.4.5.2

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

$y = \frac{\sqrt[3]{x - 6} {\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}$

Step 3.5.4.5.3

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

$y = \frac{\sqrt[3]{x - 6} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

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.

$y = \frac{\sqrt[3]{x - 6} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Step 3.5.4.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{x - 6} {\sqrt[3]{5}}^{2}}{5^{1}}$

Step 3.5.4.5.5

Evaluate the exponent.

$y = \frac{\sqrt[3]{x - 6} {\sqrt[3]{5}}^{2}}{5}$

Step 3.5.5

Simplify the numerator.

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Step 3.5.5.1

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$y = \frac{\sqrt[3]{x - 6} \sqrt[3]{5^{2}}}{5}$

Step 3.5.5.2

Raise $5$ to the power of $2$ .

$y = \frac{\sqrt[3]{x - 6} \sqrt[3]{25}}{5}$

Step 3.5.6

Simplify with factoring out.

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Step 3.5.6.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(x - 6) \cdot 25}}{5}$

Step 3.5.6.2

Reorder factors in $\frac{\sqrt[3]{(x - 6) \cdot 25}}{5}$ .

$y = \frac{\sqrt[3]{25 (x - 6)}}{5}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[3]{25 (x - 6)}}{5}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[3]{25 (x - 6)}}{5}$ is the inverse of $f (x) = 5 x^{3} + 6$ .

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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 $f^{- 1} (5 x^{3} + 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 x^{3} + 6) = \frac{\sqrt[3]{25 ((5 x^{3} + 6) - 6)}}{5}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Subtract $6$ from $6$ .

$f^{- 1} (5 x^{3} + 6) = \frac{\sqrt[3]{25 (5 x^{3} + 0)}}{5}$

Step 5.2.3.2

Add $5 x^{3}$ and $0$ .

$f^{- 1} (5 x^{3} + 6) = \frac{\sqrt[3]{25 \cdot (5 x^{3})}}{5}$

Step 5.2.3.3

Multiply $25$ by $5$ .

$f^{- 1} (5 x^{3} + 6) = \frac{\sqrt[3]{125 x^{3}}}{5}$

Step 5.2.3.4

Rewrite $125 x^{3}$ as ${(5 x)}^{3}$ .

$f^{- 1} (5 x^{3} + 6) = \frac{\sqrt[3]{{(5 x)}^{3}}}{5}$

Step 5.2.3.5

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

$f^{- 1} (5 x^{3} + 6) = \frac{5 x}{5}$

Step 5.2.4

Cancel the common factor of $5$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (5 x^{3} + 6) = \frac{5 x}{5}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (5 x^{3} + 6) = x$

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 $f (\frac{\sqrt[3]{25 (x - 6)}}{5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 {(\frac{\sqrt[3]{25 (x - 6)}}{5})}^{3} + 6$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the product rule to $\frac{\sqrt[3]{25 (x - 6)}}{5}$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{{\sqrt[3]{25 (x - 6)}}^{3}}{5^{3}}) + 6$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

Rewrite ${\sqrt[3]{25 (x - 6)}}^{3}$ as $25 (x - 6)$ .

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Step 5.3.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25 (x - 6)}$ as ${(25 (x - 6))}^{\frac{1}{3}}$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{{({(25 (x - 6))}^{\frac{1}{3}})}^{3}}{5^{3}}) + 6$

Step 5.3.3.2.1.2

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

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{{(25 (x - 6))}^{\frac{1}{3} \cdot 3}}{5^{3}}) + 6$

Step 5.3.3.2.1.3

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

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{{(25 (x - 6))}^{\frac{3}{3}}}{5^{3}}) + 6$

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.

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{{(25 (x - 6))}^{\frac{3}{3}}}{5^{3}}) + 6$

Step 5.3.3.2.1.4.2

Rewrite the expression.

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{25 (x - 6)}{5^{3}}) + 6$

Step 5.3.3.2.1.5

Simplify.

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{25 (x - 6)}{5^{3}}) + 6$

Step 5.3.3.2.2

Apply the distributive property.

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{25 x + 25 \cdot - 6}{5^{3}}) + 6$

Step 5.3.3.2.3

Multiply $25$ by $- 6$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{25 x - 150}{5^{3}}) + 6$

Step 5.3.3.2.4

Factor $25$ out of $25 x - 150$ .

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Step 5.3.3.2.4.1

Factor $25$ out of $25 x$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{25 (x) - 150}{5^{3}}) + 6$

Step 5.3.3.2.4.2

Factor $25$ out of $- 150$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{25 x + 25 \cdot - 6}{5^{3}}) + 6$

Step 5.3.3.2.4.3

Factor $25$ out of $25 x + 25 \cdot - 6$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{25 (x - 6)}{5^{3}}) + 6$

Step 5.3.3.3

Raise $5$ to the power of $3$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{25 (x - 6)}{125}) + 6$

Step 5.3.3.4

Cancel the common factor of $5$ .

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Step 5.3.3.4.1

Factor $5$ out of $125$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{25 (x - 6)}{5 (25)}) + 6$

Step 5.3.3.4.2

Cancel the common factor.

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = 5 (\frac{25 (x - 6)}{5 \cdot 25}) + 6$

Step 5.3.3.4.3

Rewrite the expression.

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = \frac{25 (x - 6)}{25} + 6$

Step 5.3.3.5

Cancel the common factor of $25$ .

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Step 5.3.3.5.1

Cancel the common factor.

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = \frac{25 (x - 6)}{25} + 6$

Step 5.3.3.5.2

Divide $x - 6$ by $1$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = x - 6 + 6$

Step 5.3.4

Combine the opposite terms in $x - 6 + 6$ .

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Step 5.3.4.1

Add $- 6$ and $6$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{\sqrt[3]{25 (x - 6)}}{5}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[3]{25 (x - 6)}}{5}$ is the inverse of $f (x) = 5 x^{3} + 6$ .

$f^{- 1} (x) = \frac{\sqrt[3]{25 (x - 6)}}{5}$

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