Precalculus Examples

$f (x) = 8 x^{3}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = 8 x^{3}$ as an equation.

$y = 8 x^{3}$

Step 2

Interchange the variables.

$x = 8 y^{3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $8 y^{3} = x$ .

$8 y^{3} = x$

Step 3.2

Divide each term in $8 y^{3} = x$ by $8$ and simplify.

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Step 3.2.1

Divide each term in $8 y^{3} = x$ by $8$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $8$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $y^{3}$ by $1$ .

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

Step 3.3

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

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

Step 3.4

Simplify $\sqrt[3]{\frac{x}{8}}$ .

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Step 3.4.1

Rewrite $\frac{x}{8}$ as ${(\frac{1}{2})}^{3} x$ .

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Step 3.4.1.1

Factor the perfect power $1^{3}$ out of $x$ .

$y = \sqrt[3]{\frac{1^{3} x}{8}}$

Step 3.4.1.2

Factor the perfect power $2^{3}$ out of $8$ .

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

Step 3.4.1.3

Rearrange the fraction $\frac{1^{3} x}{2^{3} \cdot 1}$ .

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

Step 3.4.2

Pull terms out from under the radical.

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

Step 3.4.3

Combine $\frac{1}{2}$ and $\sqrt[3]{x}$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

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

Step 5.2.3.2

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

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

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

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

Step 5.2.4.2

Divide $x$ by $1$ .

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

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

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

Step 5.3.3

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

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

Step 5.3.4

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

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Step 5.3.4.1

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

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

Step 5.3.4.2

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

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

Step 5.3.4.3

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

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

Step 5.3.4.4

Cancel the common factor of $3$ .

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Step 5.3.4.4.1

Cancel the common factor.

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

Step 5.3.4.4.2

Rewrite the expression.

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

Step 5.3.4.5

Simplify.

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

Step 5.3.5

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

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

Step 5.3.6

Cancel the common factor of $8$ .

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Step 5.3.6.1

Cancel the common factor.

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

Step 5.3.6.2

Rewrite the expression.

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

Step 5.4

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

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