Algebra Examples

$f (x) = \sqrt[3]{\frac{x + 6}{2}}$

Step 1

Write $f (x) = \sqrt[3]{\frac{x + 6}{2}}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[3]{\frac{y + 6}{2}} = x$ .

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

Step 3.2

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

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

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{y + 6}{2}}$ as ${(\frac{y + 6}{2})}^{\frac{1}{3}}$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(\frac{y + 6}{2})}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(\frac{y + 6}{2})}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.1.1.1

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

${(\frac{y + 6}{2})}^{\frac{1}{3} \cdot 3} = x^{3}$

Step 3.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${(\frac{y + 6}{2})}^{\frac{1}{3} \cdot 3} = x^{3}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

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

Step 3.3.2.1.2

Simplify.

$\frac{y + 6}{2} = x^{3}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Multiply both sides of the equation by $2$ .

$2 \frac{y + 6}{2} = 2 x^{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$2 \frac{y + 6}{2} = 2 x^{3}$

Step 3.4.2.1.2

Rewrite the expression.

$y + 6 = 2 x^{3}$

Step 3.4.3

Subtract $6$ from both sides of the equation.

$y = 2 x^{3} - 6$

Step 4

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

$f^{- 1} (x) = 2 x^{3} - 6$

Step 5

Verify if $f^{- 1} (x) = 2 x^{3} - 6$ is the inverse of $f (x) = \sqrt[3]{\frac{x + 6}{2}}$ .

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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 $f^{- 1} (\sqrt[3]{\frac{x + 6}{2}})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Rewrite $\sqrt[3]{\frac{x + 6}{2}}$ as $\frac{\sqrt[3]{x + 6}}{\sqrt[3]{2}}$ .

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

Step 5.2.3.2

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

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 {(\frac{\sqrt[3]{x + 6}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}})}^{3} - 6$

Step 5.2.3.3

Combine and simplify the denominator.

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Step 5.2.3.3.1

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

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

Step 5.2.3.3.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 5.2.3.3.3

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

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

Step 5.2.3.3.4

Add $1$ and $2$ .

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

Step 5.2.3.3.5

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

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Step 5.2.3.3.5.1

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

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

Step 5.2.3.3.5.2

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

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 {(\frac{\sqrt[3]{x + 6} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}})}^{3} - 6$

Step 5.2.3.3.5.3

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

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

Step 5.2.3.3.5.4

Cancel the common factor of $3$ .

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Step 5.2.3.3.5.4.1

Cancel the common factor.

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

Step 5.2.3.3.5.4.2

Rewrite the expression.

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

Step 5.2.3.3.5.5

Evaluate the exponent.

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

Step 5.2.3.4

Simplify the numerator.

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Step 5.2.3.4.1

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

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

Step 5.2.3.4.2

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

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 {(\frac{\sqrt[3]{x + 6} \sqrt[3]{4}}{2})}^{3} - 6$

Step 5.2.3.5

Combine using the product rule for radicals.

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 {(\frac{\sqrt[3]{(x + 6) \cdot 4}}{2})}^{3} - 6$

Step 5.2.3.6

Apply the product rule to $\frac{\sqrt[3]{(x + 6) \cdot 4}}{2}$ .

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{{\sqrt[3]{(x + 6) \cdot 4}}^{3}}{2^{3}}) - 6$

Step 5.2.3.7

Simplify the numerator.

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Step 5.2.3.7.1

Rewrite ${\sqrt[3]{(x + 6) \cdot 4}}^{3}$ as $(x + 6) \cdot 4$ .

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Step 5.2.3.7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{(x + 6) \cdot 4}$ as ${((x + 6) \cdot 4)}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{{({((x + 6) \cdot 4)}^{\frac{1}{3}})}^{3}}{2^{3}}) - 6$

Step 5.2.3.7.1.2

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

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{{((x + 6) \cdot 4)}^{\frac{1}{3} \cdot 3}}{2^{3}}) - 6$

Step 5.2.3.7.1.3

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

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{{((x + 6) \cdot 4)}^{\frac{3}{3}}}{2^{3}}) - 6$

Step 5.2.3.7.1.4

Cancel the common factor of $3$ .

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Step 5.2.3.7.1.4.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{{((x + 6) \cdot 4)}^{\frac{3}{3}}}{2^{3}}) - 6$

Step 5.2.3.7.1.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{(x + 6) \cdot 4}{2^{3}}) - 6$

Step 5.2.3.7.1.5

Simplify.

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{(x + 6) \cdot 4}{2^{3}}) - 6$

Step 5.2.3.7.2

Apply the distributive property.

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{x \cdot 4 + 6 \cdot 4}{2^{3}}) - 6$

Step 5.2.3.7.3

Move $4$ to the left of $x$ .

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{4 \cdot x + 6 \cdot 4}{2^{3}}) - 6$

Step 5.2.3.7.4

Multiply $6$ by $4$ .

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{4 \cdot x + 24}{2^{3}}) - 6$

Step 5.2.3.7.5

Factor $4$ out of $4 x + 24$ .

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Step 5.2.3.7.5.1

Factor $4$ out of $4 x$ .

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{4 (x) + 24}{2^{3}}) - 6$

Step 5.2.3.7.5.2

Factor $4$ out of $24$ .

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{4 x + 4 \cdot 6}{2^{3}}) - 6$

Step 5.2.3.7.5.3

Factor $4$ out of $4 x + 4 \cdot 6$ .

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{4 (x + 6)}{2^{3}}) - 6$

Step 5.2.3.8

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

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{4 (x + 6)}{8}) - 6$

Step 5.2.3.9

Cancel the common factor of $2$ .

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Step 5.2.3.9.1

Factor $2$ out of $8$ .

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{4 (x + 6)}{2 (4)}) - 6$

Step 5.2.3.9.2

Cancel the common factor.

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = 2 (\frac{4 (x + 6)}{2 \cdot 4}) - 6$

Step 5.2.3.9.3

Rewrite the expression.

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = \frac{4 (x + 6)}{4} - 6$

Step 5.2.3.10

Cancel the common factor of $4$ .

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Step 5.2.3.10.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = \frac{4 (x + 6)}{4} - 6$

Step 5.2.3.10.2

Divide $x + 6$ by $1$ .

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

Step 5.2.4

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

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Step 5.2.4.1

Subtract $6$ from $6$ .

$f^{- 1} (\sqrt[3]{\frac{x + 6}{2}}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

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

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 $f (2 x^{3} - 6)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (2 x^{3} - 6) = \sqrt[3]{\frac{(2 x^{3} - 6) + 6}{2}}$

Step 5.3.3

Add $- 6$ and $6$ .

$f (2 x^{3} - 6) = \sqrt[3]{\frac{2 x^{3} + 0}{2}}$

Step 5.3.4

Add $2 x^{3}$ and $0$ .

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

Step 5.3.5

Reduce the expression by cancelling the common factors.

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Step 5.3.5.1

Reduce the expression $\frac{2 x^{3}}{2}$ by cancelling the common factors.

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Step 5.3.5.1.1

Cancel the common factor.

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

Step 5.3.5.1.2

Rewrite the expression.

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

Step 5.3.5.2

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

$f (2 x^{3} - 6) = \sqrt[3]{x^{3}}$

Step 5.3.6

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

$f (2 x^{3} - 6) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 2 x^{3} - 6$ is the inverse of $f (x) = \sqrt[3]{\frac{x + 6}{2}}$ .

$f^{- 1} (x) = 2 x^{3} - 6$