Precalculus Examples

$f (x) = x^{3} - 9$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = x^{3} - 9$ as an equation.

$y = x^{3} - 9$

Step 2

Interchange the variables.

$x = y^{3} - 9$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y^{3} - 9 = x$ .

$y^{3} - 9 = x$

Step 3.2

Add $9$ to both sides of the equation.

$y^{3} = x + 9$

Step 3.3

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

$y = \sqrt[3]{x + 9}$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \sqrt[3]{x + 9}$ is the inverse of $f (x) = x^{3} - 9$ .

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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 $f^{- 1} (x^{3} - 9)$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Add $- 9$ and $9$ .

$f^{- 1} (x^{3} - 9) = \sqrt[3]{x^{3} + 0}$

Step 5.2.4

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

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

Step 5.2.5

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

$f^{- 1} (x^{3} - 9) = x$

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 $f (\sqrt[3]{x + 9})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[3]{x + 9}) = {(\sqrt[3]{x + 9})}^{3} - 9$

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

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

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

Step 5.3.3.3

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

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

Step 5.3.3.4

Cancel the common factor of $3$ .

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Step 5.3.3.4.1

Cancel the common factor.

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

Step 5.3.3.4.2

Rewrite the expression.

$f (\sqrt[3]{x + 9}) = (x + 9) - 9$

Step 5.3.3.5

Simplify.

$f (\sqrt[3]{x + 9}) = x + 9 - 9$

Step 5.3.4

Combine the opposite terms in $x + 9 - 9$ .

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Step 5.3.4.1

Subtract $9$ from $9$ .

$f (\sqrt[3]{x + 9}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\sqrt[3]{x + 9}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[3]{x + 9}$ is the inverse of $f (x) = x^{3} - 9$ .

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