Find the Inverse F(x)=x^3-5/6

$F (x) = x^{3} - \frac{5}{6}$

Step 1

Write $F (x) = x^{3} - \frac{5}{6}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

Rewrite the equation as $y^{3} - \frac{5}{6} = x$ .

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

Add $\frac{5}{6}$ to both sides of the equation.

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

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

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

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

To write $x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Simplify terms.

Combine $x$ and $\frac{6}{6}$ .

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

Combine the numerators over the common denominator.

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

Move $6$ to the left of $x$ .

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

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

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

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

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

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{6}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

Rewrite ${\sqrt[3]{6}}^{3}$ as $6$ .

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Rewrite ${\sqrt[3]{6}}^{2}$ as ${(6^{2})}^{\frac{1}{3}}$ .

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

Raise $6$ to the power of $2$ .

$y = \frac{\sqrt[3]{6 x + 5} \sqrt[3]{36}}{6}$

Simplify with factoring out.

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{(6 x + 5) \cdot 36}}{6}$

Reorder factors in $\frac{\sqrt[3]{(6 x + 5) \cdot 36}}{6}$ .

$y = \frac{\sqrt[3]{36 (6 x + 5)}}{6}$

Step 4

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

$F^{- 1} (x) = \frac{\sqrt[3]{36 (6 x + 5)}}{6}$

Step 5

Verify if $F^{- 1} (x) = \frac{\sqrt[3]{36 (6 x + 5)}}{6}$ is the inverse of $F (x) = x^{3} - \frac{5}{6}$ .

To verify the inverse, check if $F^{- 1} (F (x)) = x$ and $F (F^{- 1} (x)) = x$ .

Evaluate $F^{- 1} (F (x))$ .

Set up the composite result function.

$F^{- 1} (F (x))$

Evaluate $F^{- 1} (x^{3} - \frac{5}{6})$ by substituting in the value of $F$ into $F^{- 1}$ .

$F^{- 1} (x^{3} - \frac{5}{6}) = \frac{\sqrt[3]{36 (6 (x^{3} - \frac{5}{6}) + 5)}}{6}$

Simplify the numerator.

To write $x^{3}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$F^{- 1} (x^{3} - \frac{5}{6}) = \frac{\sqrt[3]{36 (6 (x^{3} \cdot \frac{6}{6} - \frac{5}{6}) + 5)}}{6}$

Combine $x^{3}$ and $\frac{6}{6}$ .

$F^{- 1} (x^{3} - \frac{5}{6}) = \frac{\sqrt[3]{36 (6 (\frac{x^{3} \cdot 6}{6} - \frac{5}{6}) + 5)}}{6}$

Combine the numerators over the common denominator.

$F^{- 1} (x^{3} - \frac{5}{6}) = \frac{\sqrt[3]{36 (6 (\frac{x^{3} \cdot 6 - 5}{6}) + 5)}}{6}$

Move $6$ to the left of $x^{3}$ .

$F^{- 1} (x^{3} - \frac{5}{6}) = \frac{\sqrt[3]{36 (6 (\frac{6 x^{3} - 5}{6}) + 5)}}{6}$

Cancel the common factor of $6$ .

Cancel the common factor.

$F^{- 1} (x^{3} - \frac{5}{6}) = \frac{\sqrt[3]{36 (6 (\frac{6 x^{3} - 5}{6}) + 5)}}{6}$

Rewrite the expression.

$F^{- 1} (x^{3} - \frac{5}{6}) = \frac{\sqrt[3]{36 (6 x^{3} - 5 + 5)}}{6}$

Add $- 5$ and $5$ .

$F^{- 1} (x^{3} - \frac{5}{6}) = \frac{\sqrt[3]{36 (6 x^{3} + 0)}}{6}$

Add $6 x^{3}$ and $0$ .

$F^{- 1} (x^{3} - \frac{5}{6}) = \frac{\sqrt[3]{36 \cdot (6 x^{3})}}{6}$

Multiply $36$ by $6$ .

$F^{- 1} (x^{3} - \frac{5}{6}) = \frac{\sqrt[3]{216 x^{3}}}{6}$

Rewrite $216 x^{3}$ as ${(6 x)}^{3}$ .

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

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

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

Cancel the common factor of $6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Evaluate $F (F^{- 1} (x))$ .

Set up the composite result function.

$F (F^{- 1} (x))$

Evaluate $F (\frac{\sqrt[3]{36 (6 x + 5)}}{6})$ by substituting in the value of $F^{- 1}$ into $F$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = {(\frac{\sqrt[3]{36 (6 x + 5)}}{6})}^{3} - \frac{5}{6}$

Simplify each term.

Apply the product rule to $\frac{\sqrt[3]{36 (6 x + 5)}}{6}$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{{\sqrt[3]{36 (6 x + 5)}}^{3}}{6^{3}} - \frac{5}{6}$

Simplify the numerator.

Rewrite ${\sqrt[3]{36 (6 x + 5)}}^{3}$ as $36 (6 x + 5)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{36 (6 x + 5)}$ as ${(36 (6 x + 5))}^{\frac{1}{3}}$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{{({(36 (6 x + 5))}^{\frac{1}{3}})}^{3}}{6^{3}} - \frac{5}{6}$

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

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{{(36 (6 x + 5))}^{\frac{1}{3} \cdot 3}}{6^{3}} - \frac{5}{6}$

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

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{{(36 (6 x + 5))}^{\frac{3}{3}}}{6^{3}} - \frac{5}{6}$

Cancel the common factor of $3$ .

Cancel the common factor.

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{{(36 (6 x + 5))}^{\frac{3}{3}}}{6^{3}} - \frac{5}{6}$

Rewrite the expression.

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{36 (6 x + 5)}{6^{3}} - \frac{5}{6}$

Simplify.

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{36 (6 x + 5)}{6^{3}} - \frac{5}{6}$

Apply the distributive property.

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{36 (6 x) + 36 \cdot 5}{6^{3}} - \frac{5}{6}$

Multiply $6$ by $36$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{216 x + 36 \cdot 5}{6^{3}} - \frac{5}{6}$

Multiply $36$ by $5$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{216 x + 180}{6^{3}} - \frac{5}{6}$

Factor $36$ out of $216 x + 180$ .

Factor $36$ out of $216 x$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{36 (6 x) + 180}{6^{3}} - \frac{5}{6}$

Factor $36$ out of $180$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{36 (6 x) + 36 (5)}{6^{3}} - \frac{5}{6}$

Factor $36$ out of $36 (6 x) + 36 (5)$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{36 (6 x + 5)}{6^{3}} - \frac{5}{6}$

Raise $6$ to the power of $3$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{36 (6 x + 5)}{216} - \frac{5}{6}$

Cancel the common factors.

Factor $36$ out of $216$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{36 (6 x + 5)}{36 \cdot 6} - \frac{5}{6}$

Cancel the common factor.

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{36 (6 x + 5)}{36 \cdot 6} - \frac{5}{6}$

Rewrite the expression.

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{6 x + 5}{6} - \frac{5}{6}$

Simplify terms.

Combine the numerators over the common denominator.

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{6 x + 5 - 5}{6}$

Combine the opposite terms in $6 x + 5 - 5$ .

Subtract $5$ from $5$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{6 x + 0}{6}$

Add $6 x$ and $0$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{6 x}{6}$

Cancel the common factor of $6$ .

Cancel the common factor.

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = \frac{6 x}{6}$

Divide $x$ by $1$ .

$F (\frac{\sqrt[3]{36 (6 x + 5)}}{6}) = x$

Since $F^{- 1} (F (x)) = x$ and $F (F^{- 1} (x)) = x$ , then $F^{- 1} (x) = \frac{\sqrt[3]{36 (6 x + 5)}}{6}$ is the inverse of $F (x) = x^{3} - \frac{5}{6}$ .

$F^{- 1} (x) = \frac{\sqrt[3]{36 (6 x + 5)}}{6}$

Find the Inverse F(x)=x^3-5/6

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