Precalculus Examples

$f (x) = 10 \sqrt[4]{x}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = 10 \sqrt[4]{x}$ as an equation.

$y = 10 \sqrt[4]{x}$

Step 2

Interchange the variables.

$x = 10 \sqrt[4]{y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $10 \sqrt[4]{y} = x$ .

$10 \sqrt[4]{y} = x$

Step 3.2

Divide each term in $10 \sqrt[4]{y} = x$ by $10$ and simplify.

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Step 3.2.1

Divide each term in $10 \sqrt[4]{y} = x$ by $10$ .

$\frac{10 \sqrt[4]{y}}{10} = \frac{x}{10}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $10$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{10 \sqrt[4]{y}}{10} = \frac{x}{10}$

Step 3.2.2.1.2

Divide $\sqrt[4]{y}$ by $1$ .

$\sqrt[4]{y} = \frac{x}{10}$

Step 3.3

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{y}}^{4} = {(\frac{x}{10})}^{4}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

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

${(y^{\frac{1}{4}})}^{4} = {(\frac{x}{10})}^{4}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify ${(y^{\frac{1}{4}})}^{4}$ .

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Step 3.4.2.1.1

Multiply the exponents in ${(y^{\frac{1}{4}})}^{4}$ .

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Step 3.4.2.1.1.1

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

$y^{\frac{1}{4} \cdot 4} = {(\frac{x}{10})}^{4}$

Step 3.4.2.1.1.2

Cancel the common factor of $4$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{4} \cdot 4} = {(\frac{x}{10})}^{4}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(\frac{x}{10})}^{4}$

Step 3.4.2.1.2

Simplify.

$y = {(\frac{x}{10})}^{4}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(\frac{x}{10})}^{4}$ .

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Step 3.4.3.1.1

Apply the product rule to $\frac{x}{10}$ .

$y = \frac{x^{4}}{10^{4}}$

Step 3.4.3.1.2

Raise $10$ to the power of $4$ .

$y = \frac{x^{4}}{10000}$

Step 4

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

$f^{- 1} (x) = \frac{x^{4}}{10000}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{4}}{10000}$ is the inverse of $f (x) = 10 \sqrt[4]{x}$ .

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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 $f^{- 1} (10 \sqrt[4]{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (10 \sqrt[4]{x}) = \frac{{(10 \sqrt[4]{x})}^{4}}{10000}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Apply the product rule to $10 \sqrt[4]{x}$ .

$f^{- 1} (10 \sqrt[4]{x}) = \frac{10^{4} {\sqrt[4]{x}}^{4}}{10000}$

Step 5.2.3.2

Raise $10$ to the power of $4$ .

$f^{- 1} (10 \sqrt[4]{x}) = \frac{10000 {\sqrt[4]{x}}^{4}}{10000}$

Step 5.2.3.3

Rewrite ${\sqrt[4]{x}}^{4}$ as $x$ .

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Step 5.2.3.3.1

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

$f^{- 1} (10 \sqrt[4]{x}) = \frac{10000 {(x^{\frac{1}{4}})}^{4}}{10000}$

Step 5.2.3.3.2

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

$f^{- 1} (10 \sqrt[4]{x}) = \frac{10000 x^{\frac{1}{4} \cdot 4}}{10000}$

Step 5.2.3.3.3

Combine $\frac{1}{4}$ and $4$ .

$f^{- 1} (10 \sqrt[4]{x}) = \frac{10000 x^{\frac{4}{4}}}{10000}$

Step 5.2.3.3.4

Cancel the common factor of $4$ .

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Step 5.2.3.3.4.1

Cancel the common factor.

$f^{- 1} (10 \sqrt[4]{x}) = \frac{10000 x^{\frac{4}{4}}}{10000}$

Step 5.2.3.3.4.2

Rewrite the expression.

$f^{- 1} (10 \sqrt[4]{x}) = \frac{10000 x}{10000}$

Step 5.2.3.3.5

Simplify.

$f^{- 1} (10 \sqrt[4]{x}) = \frac{10000 x}{10000}$

Step 5.2.4

Cancel the common factor of $10000$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (10 \sqrt[4]{x}) = \frac{10000 x}{10000}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (10 \sqrt[4]{x}) = x$

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 $f (\frac{x^{4}}{10000})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{4}}{10000}) = 10 \sqrt[4]{\frac{x^{4}}{10000}}$

Step 5.3.3

Remove parentheses.

$f (\frac{x^{4}}{10000}) = 10 \sqrt[4]{\frac{x^{4}}{10000}}$

Step 5.3.4

Rewrite $10000$ as $10^{4}$ .

$f (\frac{x^{4}}{10000}) = 10 \sqrt[4]{\frac{x^{4}}{10^{4}}}$

Step 5.3.5

Rewrite $\frac{x^{4}}{10^{4}}$ as ${(\frac{x}{10})}^{4}$ .

$f (\frac{x^{4}}{10000}) = 10 \sqrt[4]{{(\frac{x}{10})}^{4}}$

Step 5.3.6

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

$f (\frac{x^{4}}{10000}) = 10 (\frac{x}{10})$

Step 5.3.7

Cancel the common factor of $10$ .

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Step 5.3.7.1

Cancel the common factor.

$f (\frac{x^{4}}{10000}) = 10 (\frac{x}{10})$

Step 5.3.7.2

Rewrite the expression.

$f (\frac{x^{4}}{10000}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{4}}{10000}$ is the inverse of $f (x) = 10 \sqrt[4]{x}$ .

$f^{- 1} (x) = \frac{x^{4}}{10000}$