Precalculus Examples

$f (x) = \frac{3 x \sqrt{x}}{8}$

Step 1

Write $f (x) = \frac{3 x \sqrt{x}}{8}$ as an equation.

$y = \frac{3 x \sqrt{x}}{8}$

Step 2

Interchange the variables.

$x = \frac{3 y \sqrt{y}}{8}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{3 y \sqrt{y}}{8} = x$ .

$\frac{3 y \sqrt{y}}{8} = x$

Step 3.2

Multiply both sides of the equation by $\frac{8}{3}$ .

$\frac{8}{3} \cdot \frac{3 y \sqrt{y}}{8} = \frac{8}{3} x$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{8}{3} \cdot \frac{3 y \sqrt{y}}{8}$ .

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Step 3.3.1.1.1

Cancel the common factor of $8$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{8}{3} \cdot \frac{3 y \sqrt{y}}{8} = \frac{8}{3} x$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 y \sqrt{y}) = \frac{8}{3} x$

Step 3.3.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.1.1.2.1

Factor $3$ out of $3 y \sqrt{y}$ .

$\frac{1}{3} (3 (y \sqrt{y})) = \frac{8}{3} x$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 (y \sqrt{y})) = \frac{8}{3} x$

Step 3.3.1.1.2.3

Rewrite the expression.

$y \sqrt{y} = \frac{8}{3} x$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Combine $\frac{8}{3}$ and $x$ .

$y \sqrt{y} = \frac{8 x}{3}$

Step 3.4

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

${(y \sqrt{y})}^{2} = {(\frac{8 x}{3})}^{2}$

Step 3.5

Simplify each side of the equation.

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Step 3.5.1

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

${(y \cdot y^{\frac{1}{2}})}^{2} = {(\frac{8 x}{3})}^{2}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Simplify ${(y \cdot y^{\frac{1}{2}})}^{2}$ .

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Step 3.5.2.1.1

Multiply $y$ by $y^{\frac{1}{2}}$ by adding the exponents.

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Step 3.5.2.1.1.1

Multiply $y$ by $y^{\frac{1}{2}}$ .

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Step 3.5.2.1.1.1.1

Raise $y$ to the power of $1$ .

${(y^{1} y^{\frac{1}{2}})}^{2} = {(\frac{8 x}{3})}^{2}$

Step 3.5.2.1.1.1.2

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

${(y^{1 + \frac{1}{2}})}^{2} = {(\frac{8 x}{3})}^{2}$

Step 3.5.2.1.1.2

Write $1$ as a fraction with a common denominator.

${(y^{\frac{2}{2} + \frac{1}{2}})}^{2} = {(\frac{8 x}{3})}^{2}$

Step 3.5.2.1.1.3

Combine the numerators over the common denominator.

${(y^{\frac{2 + 1}{2}})}^{2} = {(\frac{8 x}{3})}^{2}$

Step 3.5.2.1.1.4

Add $2$ and $1$ .

${(y^{\frac{3}{2}})}^{2} = {(\frac{8 x}{3})}^{2}$

Step 3.5.2.1.2

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

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Step 3.5.2.1.2.1

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

$y^{\frac{3}{2} \cdot 2} = {(\frac{8 x}{3})}^{2}$

Step 3.5.2.1.2.2

Cancel the common factor of $2$ .

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Step 3.5.2.1.2.2.1

Cancel the common factor.

$y^{\frac{3}{2} \cdot 2} = {(\frac{8 x}{3})}^{2}$

Step 3.5.2.1.2.2.2

Rewrite the expression.

$y^{3} = {(\frac{8 x}{3})}^{2}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify ${(\frac{8 x}{3})}^{2}$ .

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Step 3.5.3.1.1

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

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Step 3.5.3.1.1.1

Apply the product rule to $\frac{8 x}{3}$ .

$y^{3} = \frac{{(8 x)}^{2}}{3^{2}}$

Step 3.5.3.1.1.2

Apply the product rule to $8 x$ .

$y^{3} = \frac{8^{2} x^{2}}{3^{2}}$

Step 3.5.3.1.2

Raise $8$ to the power of $2$ .

$y^{3} = \frac{64 x^{2}}{3^{2}}$

Step 3.5.3.1.3

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

$y^{3} = \frac{64 x^{2}}{9}$

Step 3.6

Solve for $y$ .

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Step 3.6.1

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

$y = \sqrt[3]{\frac{64 x^{2}}{9}}$

Step 3.6.2

Simplify $\sqrt[3]{\frac{64 x^{2}}{9}}$ .

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Step 3.6.2.1

Rewrite $\sqrt[3]{\frac{64 x^{2}}{9}}$ as $\frac{\sqrt[3]{64 x^{2}}}{\sqrt[3]{9}}$ .

$y = \frac{\sqrt[3]{64 x^{2}}}{\sqrt[3]{9}}$

Step 3.6.2.2

Simplify the numerator.

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Step 3.6.2.2.1

Rewrite $64$ as $4^{3}$ .

$y = \frac{\sqrt[3]{4^{3} x^{2}}}{\sqrt[3]{9}}$

Step 3.6.2.2.2

Pull terms out from under the radical.

$y = \frac{4 \sqrt[3]{x^{2}}}{\sqrt[3]{9}}$

Step 3.6.2.3

Multiply $\frac{4 \sqrt[3]{x^{2}}}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$y = \frac{4 \sqrt[3]{x^{2}}}{\sqrt[3]{9}} \cdot \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$

Step 3.6.2.4

Combine and simplify the denominator.

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Step 3.6.2.4.1

Multiply $\frac{4 \sqrt[3]{x^{2}}}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$y = \frac{4 \sqrt[3]{x^{2}} {\sqrt[3]{9}}^{2}}{\sqrt[3]{9} {\sqrt[3]{9}}^{2}}$

Step 3.6.2.4.2

Raise $\sqrt[3]{9}$ to the power of $1$ .

$y = \frac{4 \sqrt[3]{x^{2}} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1} {\sqrt[3]{9}}^{2}}$

Step 3.6.2.4.3

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

$y = \frac{4 \sqrt[3]{x^{2}} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1 + 2}}$

Step 3.6.2.4.4

Add $1$ and $2$ .

$y = \frac{4 \sqrt[3]{x^{2}} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{3}}$

Step 3.6.2.4.5

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

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Step 3.6.2.4.5.1

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

$y = \frac{4 \sqrt[3]{x^{2}} {\sqrt[3]{9}}^{2}}{{(9^{\frac{1}{3}})}^{3}}$

Step 3.6.2.4.5.2

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

$y = \frac{4 \sqrt[3]{x^{2}} {\sqrt[3]{9}}^{2}}{9^{\frac{1}{3} \cdot 3}}$

Step 3.6.2.4.5.3

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

$y = \frac{4 \sqrt[3]{x^{2}} {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 3.6.2.4.5.4

Cancel the common factor of $3$ .

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Step 3.6.2.4.5.4.1

Cancel the common factor.

$y = \frac{4 \sqrt[3]{x^{2}} {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 3.6.2.4.5.4.2

Rewrite the expression.

$y = \frac{4 \sqrt[3]{x^{2}} {\sqrt[3]{9}}^{2}}{9^{1}}$

Step 3.6.2.4.5.5

Evaluate the exponent.

$y = \frac{4 \sqrt[3]{x^{2}} {\sqrt[3]{9}}^{2}}{9}$

Step 3.6.2.5

Simplify the numerator.

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Step 3.6.2.5.1

Rewrite ${\sqrt[3]{9}}^{2}$ as $\sqrt[3]{9^{2}}$ .

$y = \frac{4 \sqrt[3]{x^{2}} \sqrt[3]{9^{2}}}{9}$

Step 3.6.2.5.2

Raise $9$ to the power of $2$ .

$y = \frac{4 \sqrt[3]{x^{2}} \sqrt[3]{81}}{9}$

Step 3.6.2.5.3

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 3.6.2.5.3.1

Factor $27$ out of $81$ .

$y = \frac{4 \sqrt[3]{x^{2}} \sqrt[3]{27 (3)}}{9}$

Step 3.6.2.5.3.2

Rewrite $27$ as $3^{3}$ .

$y = \frac{4 \sqrt[3]{x^{2}} \sqrt[3]{3^{3} \cdot 3}}{9}$

Step 3.6.2.5.4

Pull terms out from under the radical.

$y = \frac{4 \sqrt[3]{x^{2}} \cdot 3 \sqrt[3]{3}}{9}$

Step 3.6.2.5.5

Combine exponents.

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Step 3.6.2.5.5.1

Multiply $3$ by $4$ .

$y = \frac{12 \sqrt[3]{x^{2}} \sqrt[3]{3}}{9}$

Step 3.6.2.5.5.2

Combine using the product rule for radicals.

$y = \frac{12 \sqrt[3]{3 x^{2}}}{9}$

Step 3.6.2.6

Cancel the common factor of $12$ and $9$ .

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Step 3.6.2.6.1

Factor $3$ out of $12 \sqrt[3]{3 x^{2}}$ .

$y = \frac{3 (4 \sqrt[3]{3 x^{2}})}{9}$

Step 3.6.2.6.2

Cancel the common factors.

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Step 3.6.2.6.2.1

Factor $3$ out of $9$ .

$y = \frac{3 (4 \sqrt[3]{3 x^{2}})}{3 (3)}$

Step 3.6.2.6.2.2

Cancel the common factor.

$y = \frac{3 (4 \sqrt[3]{3 x^{2}})}{3 \cdot 3}$

Step 3.6.2.6.2.3

Rewrite the expression.

$y = \frac{4 \sqrt[3]{3 x^{2}}}{3}$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{4 \sqrt[3]{3 x^{2}}}{3}$ is the inverse of $f (x) = \frac{3 x \sqrt{x}}{8}$ .

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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 $f^{- 1} (\frac{3 x \sqrt{x}}{8})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Apply the product rule to $\frac{3 x \sqrt{x}}{8}$ .

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

Step 5.2.3.2

Simplify the numerator.

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Step 5.2.3.2.1

Apply the product rule to $3 x \sqrt{x}$ .

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

Step 5.2.3.2.2

Apply the product rule to $3 x$ .

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

Step 5.2.3.2.3

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

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 x^{2} {\sqrt{x}}^{2}}{8^{2}})}}{3}$

Step 5.2.3.2.4

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 5.2.3.2.4.1

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

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 x^{2} {(x^{\frac{1}{2}})}^{2}}{8^{2}})}}{3}$

Step 5.2.3.2.4.2

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

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 x^{2} x^{\frac{1}{2} \cdot 2}}{8^{2}})}}{3}$

Step 5.2.3.2.4.3

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 x^{2} x^{\frac{2}{2}}}{8^{2}})}}{3}$

Step 5.2.3.2.4.4

Cancel the common factor of $2$ .

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Step 5.2.3.2.4.4.1

Cancel the common factor.

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 x^{2} x^{\frac{2}{2}}}{8^{2}})}}{3}$

Step 5.2.3.2.4.4.2

Rewrite the expression.

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 x^{2} x}{8^{2}})}}{3}$

Step 5.2.3.2.4.5

Simplify.

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 x^{2} x}{8^{2}})}}{3}$

Step 5.2.3.2.5

Combine exponents.

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Step 5.2.3.2.5.1

Raise $x$ to the power of $1$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 (x \cdot x^{2})}{8^{2}})}}{3}$

Step 5.2.3.2.5.2

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

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 x^{1 + 2}}{8^{2}})}}{3}$

Step 5.2.3.2.5.3

Add $1$ and $2$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 x^{3}}{8^{2}})}}{3}$

Step 5.2.3.3

Raise $8$ to the power of $2$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{3 (\frac{9 x^{3}}{64})}}{3}$

Step 5.2.3.4

Combine $3$ and $\frac{9 x^{3}}{64}$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{\frac{3 (9 x^{3})}{64}}}{3}$

Step 5.2.3.5

Multiply $3$ by $9$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{\frac{27 x^{3}}{64}}}{3}$

Step 5.2.3.6

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

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 \sqrt[3]{\frac{{(3 x)}^{3}}{64}}}{3}$

Step 5.2.3.7

Rewrite $64$ as $4^{3}$ .

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

Step 5.2.3.8

Rewrite $\frac{{(3 x)}^{3}}{4^{3}}$ as ${(\frac{3 x}{4})}^{3}$ .

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

Step 5.2.3.9

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

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{4 (\frac{3 x}{4})}{3}$

Step 5.2.4

Combine $4$ and $\frac{3 x}{4}$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{\frac{4 (3 x)}{4}}{3}$

Step 5.2.5

Multiply $4$ by $3$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{\frac{12 x}{4}}{3}$

Step 5.2.6

Reduce the expression by cancelling the common factors.

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Step 5.2.6.1

Reduce the expression $\frac{12 x}{4}$ by cancelling the common factors.

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Step 5.2.6.1.1

Factor $4$ out of $12 x$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{\frac{4 (3 x)}{4}}{3}$

Step 5.2.6.1.2

Factor $4$ out of $4$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{\frac{4 (3 x)}{4 (1)}}{3}$

Step 5.2.6.1.3

Cancel the common factor.

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{\frac{4 (3 x)}{4 \cdot 1}}{3}$

Step 5.2.6.1.4

Rewrite the expression.

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{\frac{3 x}{1}}{3}$

Step 5.2.6.2

Divide $3 x$ by $1$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{3 x}{3}$

Step 5.2.7

Cancel the common factor of $3$ .

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Step 5.2.7.1

Cancel the common factor.

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = \frac{3 x}{3}$

Step 5.2.7.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{3 x \sqrt{x}}{8}) = x$

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 $f (\frac{4 \sqrt[3]{3 x^{2}}}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Combine $3$ and $\frac{4 \sqrt[3]{3 x^{2}}}{3}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{3 (4 \sqrt[3]{3 x^{2}})}{3} \cdot \sqrt{\frac{4 \sqrt[3]{3 x^{2}}}{3}}}{8}$

Step 5.3.3.2

Combine $\frac{3 (4 \sqrt[3]{3 x^{2}})}{3}$ and $\sqrt{\frac{4 \sqrt[3]{3 x^{2}}}{3}}$ .

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

Step 5.3.4

Simplify the numerator.

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Step 5.3.4.1

Multiply $3$ by $4$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[3]{3 x^{2}} \sqrt{\frac{4 \sqrt[3]{3 x^{2}}}{3}}}{3}}{8}$

Step 5.3.4.2

Rewrite the expression using the least common index of $6$ .

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Step 5.3.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{4 \sqrt[3]{3 x^{2}}}{3}}$ as ${(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{\frac{1}{2}}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 ({(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{\frac{1}{2}} \sqrt[3]{3 x^{2}})}{3}}{8}$

Step 5.3.4.2.2

Rewrite ${(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{\frac{1}{2}}$ as ${(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{\frac{3}{6}}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 ({(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{\frac{3}{6}} \sqrt[3]{3 x^{2}})}{3}}{8}$

Step 5.3.4.2.3

Rewrite ${(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{\frac{3}{6}}$ as $\sqrt[6]{{(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{3}}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 (\sqrt[6]{{(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{3}} \sqrt[3]{3 x^{2}})}{3}}{8}$

Step 5.3.4.2.4

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

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 (\sqrt[6]{{(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{3}} {(3 x^{2})}^{\frac{1}{3}})}{3}}{8}$

Step 5.3.4.2.5

Rewrite ${(3 x^{2})}^{\frac{1}{3}}$ as ${(3 x^{2})}^{\frac{2}{6}}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 (\sqrt[6]{{(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{3}} {(3 x^{2})}^{\frac{2}{6}})}{3}}{8}$

Step 5.3.4.2.6

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

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 (\sqrt[6]{{(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{3}} \sqrt[6]{{(3 x^{2})}^{2}})}{3}}{8}$

Step 5.3.4.3

Combine using the product rule for radicals.

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{{(\frac{4 \sqrt[3]{3 x^{2}}}{3})}^{3} {(3 x^{2})}^{2}}}{3}}{8}$

Step 5.3.4.4

Apply the product rule to $\frac{4 \sqrt[3]{3 x^{2}}}{3}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{{(4 \sqrt[3]{3 x^{2}})}^{3}}{3^{3}} \cdot {(3 x^{2})}^{2}}}{3}}{8}$

Step 5.3.4.5

Apply the product rule to $3 x^{2}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{{(4 \sqrt[3]{3 x^{2}})}^{3}}{3^{3}} \cdot (3^{2} {(x^{2})}^{2})}}{3}}{8}$

Step 5.3.4.6

Combine exponents.

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Step 5.3.4.6.1

Combine $\frac{{(4 \sqrt[3]{3 x^{2}})}^{3}}{3^{3}}$ and $3^{2}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{{(4 \sqrt[3]{3 x^{2}})}^{3} \cdot 3^{2}}{3^{3}} \cdot {(x^{2})}^{2}}}{3}}{8}$

Step 5.3.4.6.2

Combine $\frac{{(4 \sqrt[3]{3 x^{2}})}^{3} \cdot 3^{2}}{3^{3}}$ and ${(x^{2})}^{2}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{{(4 \sqrt[3]{3 x^{2}})}^{3} \cdot (3^{2} {(x^{2})}^{2})}{3^{3}}}}{3}}{8}$

Step 5.3.4.7

Cancel the common factor of $3^{2}$ and $3^{3}$ .

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Step 5.3.4.7.1

Factor $3^{2}$ out of ${(4 \sqrt[3]{3 x^{2}})}^{3} \cdot 3^{2} {(x^{2})}^{2}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{3^{2} ({(4 \sqrt[3]{3 x^{2}})}^{3} {(x^{2})}^{2})}{3^{3}}}}{3}}{8}$

Step 5.3.4.7.2

Cancel the common factors.

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Step 5.3.4.7.2.1

Factor $3^{2}$ out of $3^{3}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{3^{2} ({(4 \sqrt[3]{3 x^{2}})}^{3} {(x^{2})}^{2})}{3^{2} \cdot 3}}}{3}}{8}$

Step 5.3.4.7.2.2

Cancel the common factor.

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{3^{2} ({(4 \sqrt[3]{3 x^{2}})}^{3} {(x^{2})}^{2})}{3^{2} \cdot 3}}}{3}}{8}$

Step 5.3.4.7.2.3

Rewrite the expression.

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{{(4 \sqrt[3]{3 x^{2}})}^{3} {(x^{2})}^{2}}{3}}}{3}}{8}$

Step 5.3.4.8

Simplify the numerator.

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Step 5.3.4.8.1

Apply the product rule to $4 \sqrt[3]{3 x^{2}}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{4^{3} {\sqrt[3]{3 x^{2}}}^{3} {(x^{2})}^{2}}{3}}}{3}}{8}$

Step 5.3.4.8.2

Raise $4$ to the power of $3$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 {\sqrt[3]{3 x^{2}}}^{3} {(x^{2})}^{2}}{3}}}{3}}{8}$

Step 5.3.4.8.3

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

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Step 5.3.4.8.3.1

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

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 {({(3 x^{2})}^{\frac{1}{3}})}^{3} {(x^{2})}^{2}}{3}}}{3}}{8}$

Step 5.3.4.8.3.2

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

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 {(3 x^{2})}^{\frac{1}{3} \cdot 3} {(x^{2})}^{2}}{3}}}{3}}{8}$

Step 5.3.4.8.3.3

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

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 {(3 x^{2})}^{\frac{3}{3}} {(x^{2})}^{2}}{3}}}{3}}{8}$

Step 5.3.4.8.3.4

Cancel the common factor of $3$ .

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Step 5.3.4.8.3.4.1

Cancel the common factor.

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 {(3 x^{2})}^{\frac{3}{3}} {(x^{2})}^{2}}{3}}}{3}}{8}$

Step 5.3.4.8.3.4.2

Rewrite the expression.

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 (3 x^{2}) {(x^{2})}^{2}}{3}}}{3}}{8}$

Step 5.3.4.8.3.5

Simplify.

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 (3 x^{2}) {(x^{2})}^{2}}{3}}}{3}}{8}$

Step 5.3.4.8.4

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 5.3.4.8.4.1

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

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 (3 x^{2}) x^{2 \cdot 2}}{3}}}{3}}{8}$

Step 5.3.4.8.4.2

Multiply $2$ by $2$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 (3 x^{2}) x^{4}}{3}}}{3}}{8}$

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 \cdot (3 x^{2} x^{4})}{3}}}{3}}{8}$

Step 5.3.4.8.5

Combine exponents.

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Step 5.3.4.8.5.1

Multiply $64$ by $3$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{192 x^{2} x^{4}}{3}}}{3}}{8}$

Step 5.3.4.8.5.2

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 5.3.4.8.5.2.1

Move $x^{4}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{192 (x^{4} x^{2})}{3}}}{3}}{8}$

Step 5.3.4.8.5.2.2

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

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{192 x^{4 + 2}}{3}}}{3}}{8}$

Step 5.3.4.8.5.2.3

Add $4$ and $2$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{192 x^{6}}{3}}}{3}}{8}$

Step 5.3.4.9

Cancel the common factor of $192$ and $3$ .

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Step 5.3.4.9.1

Factor $3$ out of $192 x^{6}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{3 (64 x^{6})}{3}}}{3}}{8}$

Step 5.3.4.9.2

Cancel the common factors.

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Step 5.3.4.9.2.1

Factor $3$ out of $3$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{3 (64 x^{6})}{3 (1)}}}{3}}{8}$

Step 5.3.4.9.2.2

Cancel the common factor.

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{3 (64 x^{6})}{3 \cdot 1}}}{3}}{8}$

Step 5.3.4.9.2.3

Rewrite the expression.

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{\frac{64 x^{6}}{1}}}{3}}{8}$

Step 5.3.4.9.2.4

Divide $64 x^{6}$ by $1$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{64 x^{6}}}{3}}{8}$

Step 5.3.5

Simplify the numerator.

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Step 5.3.5.1

Rewrite $64 x^{6}$ as ${(2 x)}^{6}$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \sqrt[6]{{(2 x)}^{6}}}{3}}{8}$

Step 5.3.5.2

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

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{12 \cdot (2 x)}{3}}{8}$

Step 5.3.5.3

Multiply $12$ by $2$ .

$f (\frac{4 \sqrt[3]{3 x^{2}}}{3}) = \frac{\frac{24 x}{3}}{8}$

Step 5.3.6

Reduce the expression by cancelling the common factors.

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Step 5.3.6.1

Reduce the expression $\frac{24 x}{3}$ by cancelling the common factors.

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Step 5.3.6.1.1

Factor $3$ out of $24 x$ .

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

Step 5.3.6.1.2

Factor $3$ out of $3$ .

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

Step 5.3.6.1.3

Cancel the common factor.

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

Step 5.3.6.1.4

Rewrite the expression.

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

Step 5.3.6.2

Divide $8 x$ by $1$ .

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

Step 5.3.7

Cancel the common factor of $8$ .

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Step 5.3.7.1

Cancel the common factor.

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

Step 5.3.7.2

Divide $x$ by $1$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{4 \sqrt[3]{3 x^{2}}}{3}$ is the inverse of $f (x) = \frac{3 x \sqrt{x}}{8}$ .

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