Precalculus Examples

$f (x) = {(\frac{\sqrt[3]{x}}{10})}^{5}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = {(\frac{\sqrt[3]{x}}{10})}^{5}$ as an equation.

$y = {(\frac{\sqrt[3]{x}}{10})}^{5}$

Step 2

Interchange the variables.

$x = {(\frac{\sqrt[3]{y}}{10})}^{5}$

Step 3

Solve for $y$ .

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Step 3.1

Simplify ${(\frac{\sqrt[3]{y}}{10})}^{5}$ .

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Step 3.1.1

Apply the product rule to $\frac{\sqrt[3]{y}}{10}$ .

$x = \frac{{\sqrt[3]{y}}^{5}}{10^{5}}$

Step 3.1.2

Simplify the numerator.

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Step 3.1.2.1

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

$x = \frac{\sqrt[3]{y^{5}}}{10^{5}}$

Step 3.1.2.2

Factor out $y^{3}$ .

$x = \frac{\sqrt[3]{y^{3} y^{2}}}{10^{5}}$

Step 3.1.2.3

Pull terms out from under the radical.

$x = \frac{y \sqrt[3]{y^{2}}}{10^{5}}$

Step 3.1.3

Raise $10$ to the power of $5$ .

$x = \frac{y \sqrt[3]{y^{2}}}{100000}$

Step 3.2

Rewrite the equation as $\frac{y \sqrt[3]{y^{2}}}{100000} = x$ .

$\frac{y \sqrt[3]{y^{2}}}{100000} = x$

Step 3.3

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

$\frac{y \cdot y^{\frac{2}{3}}}{100000} = x$

Step 3.4

Multiply both sides of the equation by $100000$ .

$100000 \frac{y \cdot y^{\frac{2}{3}}}{100000} = 100000 x$

Step 3.5

Simplify the left side.

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Step 3.5.1

Simplify $100000 \frac{y \cdot y^{\frac{2}{3}}}{100000}$ .

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Step 3.5.1.1

Cancel the common factor of $100000$ .

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Step 3.5.1.1.1

Cancel the common factor.

$100000 \frac{y \cdot y^{\frac{2}{3}}}{100000} = 100000 x$

Step 3.5.1.1.2

Rewrite the expression.

$y \cdot y^{\frac{2}{3}} = 100000 x$

Step 3.5.1.2

Multiply $y$ by $y^{\frac{2}{3}}$ by adding the exponents.

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Step 3.5.1.2.1

Multiply $y$ by $y^{\frac{2}{3}}$ .

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Step 3.5.1.2.1.1

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

$y^{1} y^{\frac{2}{3}} = 100000 x$

Step 3.5.1.2.1.2

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

$y^{1 + \frac{2}{3}} = 100000 x$

Step 3.5.1.2.2

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

$y^{\frac{3}{3} + \frac{2}{3}} = 100000 x$

Step 3.5.1.2.3

Combine the numerators over the common denominator.

$y^{\frac{3 + 2}{3}} = 100000 x$

Step 3.5.1.2.4

Add $3$ and $2$ .

$y^{\frac{5}{3}} = 100000 x$

Step 3.6

Raise each side of the equation to the power of $\frac{3}{5}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{5}{3}})}^{\frac{3}{5}} = {(100000 x)}^{\frac{3}{5}}$

Step 3.7

Simplify the exponent.

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Step 3.7.1

Simplify the left side.

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Step 3.7.1.1

Simplify ${(y^{\frac{5}{3}})}^{\frac{3}{5}}$ .

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Step 3.7.1.1.1

Multiply the exponents in ${(y^{\frac{5}{3}})}^{\frac{3}{5}}$ .

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Step 3.7.1.1.1.1

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

$y^{\frac{5}{3} \cdot \frac{3}{5}} = {(100000 x)}^{\frac{3}{5}}$

Step 3.7.1.1.1.2

Cancel the common factor of $5$ .

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Step 3.7.1.1.1.2.1

Cancel the common factor.

$y^{\frac{5}{3} \cdot \frac{3}{5}} = {(100000 x)}^{\frac{3}{5}}$

Step 3.7.1.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{3} \cdot 3} = {(100000 x)}^{\frac{3}{5}}$

Step 3.7.1.1.1.3

Cancel the common factor of $3$ .

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Step 3.7.1.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(100000 x)}^{\frac{3}{5}}$

Step 3.7.1.1.1.3.2

Rewrite the expression.

$y^{1} = {(100000 x)}^{\frac{3}{5}}$

Step 3.7.1.1.2

Simplify.

$y = {(100000 x)}^{\frac{3}{5}}$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify ${(100000 x)}^{\frac{3}{5}}$ .

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Step 3.7.2.1.1

Simplify the expression.

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Step 3.7.2.1.1.1

Apply the product rule to $100000 x$ .

$y = 100000^{\frac{3}{5}} x^{\frac{3}{5}}$

Step 3.7.2.1.1.2

Rewrite $100000$ as $10^{5}$ .

$y = {(10^{5})}^{\frac{3}{5}} x^{\frac{3}{5}}$

Step 3.7.2.1.1.3

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

$y = 10^{5 (\frac{3}{5})} x^{\frac{3}{5}}$

Step 3.7.2.1.2

Cancel the common factor of $5$ .

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Step 3.7.2.1.2.1

Cancel the common factor.

$y = 10^{5 (\frac{3}{5})} x^{\frac{3}{5}}$

Step 3.7.2.1.2.2

Rewrite the expression.

$y = 10^{3} x^{\frac{3}{5}}$

Step 3.7.2.1.3

Raise $10$ to the power of $3$ .

$y = 1000 x^{\frac{3}{5}}$

Step 4

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

$f^{- 1} (x) = 1000 x^{\frac{3}{5}}$

Step 5

Verify if $f^{- 1} (x) = 1000 x^{\frac{3}{5}}$ is the inverse of $f (x) = {(\frac{\sqrt[3]{x}}{10})}^{5}$ .

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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 $f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 {({(\frac{\sqrt[3]{x}}{10})}^{5})}^{\frac{3}{5}}$

Step 5.2.3

Apply basic rules of exponents.

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Step 5.2.3.1

Multiply the exponents in ${({(\frac{\sqrt[3]{x}}{10})}^{5})}^{\frac{3}{5}}$ .

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Step 5.2.3.1.1

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

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 {(\frac{\sqrt[3]{x}}{10})}^{5 (\frac{3}{5})}$

Step 5.2.3.1.2

Cancel the common factor of $5$ .

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Step 5.2.3.1.2.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 {(\frac{\sqrt[3]{x}}{10})}^{5 (\frac{3}{5})}$

Step 5.2.3.1.2.2

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 {(\frac{\sqrt[3]{x}}{10})}^{3}$

Step 5.2.3.2

Apply the product rule to $\frac{\sqrt[3]{x}}{10}$ .

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 (\frac{{\sqrt[3]{x}}^{3}}{10^{3}})$

Step 5.2.4

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

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Step 5.2.4.1

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

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 (\frac{{(x^{\frac{1}{3}})}^{3}}{10^{3}})$

Step 5.2.4.2

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

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 (\frac{x^{\frac{1}{3} \cdot 3}}{10^{3}})$

Step 5.2.4.3

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

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 (\frac{x^{\frac{3}{3}}}{10^{3}})$

Step 5.2.4.4

Cancel the common factor of $3$ .

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Step 5.2.4.4.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 (\frac{x^{\frac{3}{3}}}{10^{3}})$

Step 5.2.4.4.2

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 (\frac{x}{10^{3}})$

Step 5.2.4.5

Simplify.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 (\frac{x}{10^{3}})$

Step 5.2.5

Raise $10$ to the power of $3$ .

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 (\frac{x}{1000})$

Step 5.2.6

Cancel the common factor of $1000$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = 1000 (\frac{x}{1000})$

Step 5.2.6.2

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{10})}^{5}) = x$

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 $f (1000 x^{\frac{3}{5}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (1000 x^{\frac{3}{5}}) = {(\frac{\sqrt[3]{1000 x^{\frac{3}{5}}}}{10})}^{5}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Rewrite $1000 x^{\frac{3}{5}}$ as ${(10 x^{\frac{1}{5}})}^{3}$ .

$f (1000 x^{\frac{3}{5}}) = {(\frac{\sqrt[3]{{(10 x^{\frac{1}{5}})}^{3}}}{10})}^{5}$

Step 5.3.3.2

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

$f (1000 x^{\frac{3}{5}}) = {(\frac{10 x^{\frac{1}{5}}}{10})}^{5}$

Step 5.3.4

Reduce the expression by cancelling the common factors.

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Step 5.3.4.1

Cancel the common factor.

$f (1000 x^{\frac{3}{5}}) = {(\frac{10 x^{\frac{1}{5}}}{10})}^{5}$

Step 5.3.4.2

Simplify the expression.

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Step 5.3.4.2.1

Divide $x^{\frac{1}{5}}$ by $1$ .

$f (1000 x^{\frac{3}{5}}) = {(x^{\frac{1}{5}})}^{5}$

Step 5.3.4.2.2

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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Step 5.3.4.2.2.1

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

$f (1000 x^{\frac{3}{5}}) = x^{\frac{1}{5} \cdot 5}$

Step 5.3.4.2.2.2

Cancel the common factor of $5$ .

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Step 5.3.4.2.2.2.1

Cancel the common factor.

$f (1000 x^{\frac{3}{5}}) = x^{\frac{1}{5} \cdot 5}$

Step 5.3.4.2.2.2.2

Rewrite the expression.

$f (1000 x^{\frac{3}{5}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 1000 x^{\frac{3}{5}}$ is the inverse of $f (x) = {(\frac{\sqrt[3]{x}}{10})}^{5}$ .

$f^{- 1} (x) = 1000 x^{\frac{3}{5}}$