Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Multiply both sides of the equation by $3$ .

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

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $3$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

$y^{5} = 3 x$

Step 3.4

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

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Combine $3$ and $\frac{x^{5}}{3}$ .

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

Step 5.2.4

Reduce the expression by cancelling the common factors.

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Step 5.2.4.1

Reduce the expression $\frac{3 x^{5}}{3}$ by cancelling the common factors.

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Step 5.2.4.1.1

Cancel the common factor.

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

Step 5.2.4.1.2

Rewrite the expression.

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

Step 5.2.4.2

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

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

Step 5.2.5

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

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

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

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

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

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

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

Step 5.3.3.3

Combine $\frac{1}{5}$ and $5$ .

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

Step 5.3.3.4

Cancel the common factor of $5$ .

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Step 5.3.3.4.1

Cancel the common factor.

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

Step 5.3.3.4.2

Rewrite the expression.

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

Step 5.3.3.5

Simplify.

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

Step 5.3.4

Cancel the common factor of $3$ .

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Step 5.3.4.1

Cancel the common factor.

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

Step 5.3.4.2

Divide $x$ by $1$ .

$f (\sqrt[5]{3 x}) = x$

Step 5.4

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

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