Precalculus Examples

$f (x) = \frac{\sqrt[3]{x + 4}}{7}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \frac{\sqrt[3]{x + 4}}{7}$ as an equation.

$y = \frac{\sqrt[3]{x + 4}}{7}$

Step 2

Interchange the variables.

$x = \frac{\sqrt[3]{y + 4}}{7}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{\sqrt[3]{y + 4}}{7} = x$ .

$\frac{\sqrt[3]{y + 4}}{7} = x$

Step 3.2

Multiply both sides of the equation by $7$ .

$7 \frac{\sqrt[3]{y + 4}}{7} = 7 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $7$ .

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Step 3.3.1.1

Cancel the common factor.

$7 \frac{\sqrt[3]{y + 4}}{7} = 7 x$

Step 3.3.1.2

Rewrite the expression.

$\sqrt[3]{y + 4} = 7 x$

Step 3.4

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{y + 4}}^{3} = {(7 x)}^{3}$

Step 3.5

Simplify each side of the equation.

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y + 4}$ as ${(y + 4)}^{\frac{1}{3}}$ .

${({(y + 4)}^{\frac{1}{3}})}^{3} = {(7 x)}^{3}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Simplify ${({(y + 4)}^{\frac{1}{3}})}^{3}$ .

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Step 3.5.2.1.1

Multiply the exponents in ${({(y + 4)}^{\frac{1}{3}})}^{3}$ .

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Step 3.5.2.1.1.1

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

${(y + 4)}^{\frac{1}{3} \cdot 3} = {(7 x)}^{3}$

Step 3.5.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.5.2.1.1.2.1

Cancel the common factor.

${(y + 4)}^{\frac{1}{3} \cdot 3} = {(7 x)}^{3}$

Step 3.5.2.1.1.2.2

Rewrite the expression.

${(y + 4)}^{1} = {(7 x)}^{3}$

Step 3.5.2.1.2

Simplify.

$y + 4 = {(7 x)}^{3}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify ${(7 x)}^{3}$ .

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Step 3.5.3.1.1

Apply the product rule to $7 x$ .

$y + 4 = 7^{3} x^{3}$

Step 3.5.3.1.2

Raise $7$ to the power of $3$ .

$y + 4 = 343 x^{3}$

Step 3.6

Subtract $4$ from both sides of the equation.

$y = 343 x^{3} - 4$

Step 4

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

$f^{- 1} (x) = 343 x^{3} - 4$

Step 5

Verify if $f^{- 1} (x) = 343 x^{3} - 4$ is the inverse of $f (x) = \frac{\sqrt[3]{x + 4}}{7}$ .

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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 $f^{- 1} (\frac{\sqrt[3]{x + 4}}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = 343 {(\frac{\sqrt[3]{x + 4}}{7})}^{3} - 4$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Apply the product rule to $\frac{\sqrt[3]{x + 4}}{7}$ .

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = 343 (\frac{{\sqrt[3]{x + 4}}^{3}}{7^{3}}) - 4$

Step 5.2.3.2

Rewrite ${\sqrt[3]{x + 4}}^{3}$ as $x + 4$ .

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Step 5.2.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + 4}$ as ${(x + 4)}^{\frac{1}{3}}$ .

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = 343 (\frac{{({(x + 4)}^{\frac{1}{3}})}^{3}}{7^{3}}) - 4$

Step 5.2.3.2.2

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

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = 343 (\frac{{(x + 4)}^{\frac{1}{3} \cdot 3}}{7^{3}}) - 4$

Step 5.2.3.2.3

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

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = 343 (\frac{{(x + 4)}^{\frac{3}{3}}}{7^{3}}) - 4$

Step 5.2.3.2.4

Cancel the common factor of $3$ .

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Step 5.2.3.2.4.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = 343 (\frac{{(x + 4)}^{\frac{3}{3}}}{7^{3}}) - 4$

Step 5.2.3.2.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = 343 (\frac{x + 4}{7^{3}}) - 4$

Step 5.2.3.2.5

Simplify.

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = 343 (\frac{x + 4}{7^{3}}) - 4$

Step 5.2.3.3

Raise $7$ to the power of $3$ .

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = 343 (\frac{x + 4}{343}) - 4$

Step 5.2.3.4

Cancel the common factor of $343$ .

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Step 5.2.3.4.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = 343 (\frac{x + 4}{343}) - 4$

Step 5.2.3.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = x + 4 - 4$

Step 5.2.4

Combine the opposite terms in $x + 4 - 4$ .

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Step 5.2.4.1

Subtract $4$ from $4$ .

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[3]{x + 4}}{7}) = x$

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 $f (343 x^{3} - 4)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (343 x^{3} - 4) = \frac{\sqrt[3]{(343 x^{3} - 4) + 4}}{7}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Add $- 4$ and $4$ .

$f (343 x^{3} - 4) = \frac{\sqrt[3]{343 x^{3} + 0}}{7}$

Step 5.3.3.2

Add $343 x^{3}$ and $0$ .

$f (343 x^{3} - 4) = \frac{\sqrt[3]{343 x^{3}}}{7}$

Step 5.3.3.3

Rewrite $343 x^{3}$ as ${(7 x)}^{3}$ .

$f (343 x^{3} - 4) = \frac{\sqrt[3]{{(7 x)}^{3}}}{7}$

Step 5.3.3.4

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

$f (343 x^{3} - 4) = \frac{7 x}{7}$

Step 5.3.4

Cancel the common factor of $7$ .

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Step 5.3.4.1

Cancel the common factor.

$f (343 x^{3} - 4) = \frac{7 x}{7}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (343 x^{3} - 4) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 343 x^{3} - 4$ is the inverse of $f (x) = \frac{\sqrt[3]{x + 4}}{7}$ .

$f^{- 1} (x) = 343 x^{3} - 4$