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

Simplify ${(\frac{\sqrt[3]{y}}{4})}^{7}$ .

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Step 3.1.1

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

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

Step 3.1.2

Simplify the numerator.

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Step 3.1.2.1

Rewrite ${\sqrt[3]{y}}^{7}$ as $\sqrt[3]{y^{7}}$ .

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

Step 3.1.2.2

Rewrite $y^{7}$ as ${(y^{2})}^{3} y$ .

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Step 3.1.2.2.1

Factor out $y^{6}$ .

$x = \frac{\sqrt[3]{y^{6} y}}{4^{7}}$

Step 3.1.2.2.2

Rewrite $y^{6}$ as ${(y^{2})}^{3}$ .

$x = \frac{\sqrt[3]{{(y^{2})}^{3} y}}{4^{7}}$

Step 3.1.2.3

Pull terms out from under the radical.

$x = \frac{y^{2} \sqrt[3]{y}}{4^{7}}$

Step 3.1.3

Raise $4$ to the power of $7$ .

$x = \frac{y^{2} \sqrt[3]{y}}{16384}$

Step 3.2

Rewrite the equation as $\frac{y^{2} \sqrt[3]{y}}{16384} = x$ .

$\frac{y^{2} \sqrt[3]{y}}{16384} = x$

Step 3.3

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

$\frac{y^{2} y^{\frac{1}{3}}}{16384} = x$

Step 3.4

Multiply both sides of the equation by $16384$ .

$16384 \frac{y^{2} y^{\frac{1}{3}}}{16384} = 16384 x$

Step 3.5

Simplify the left side.

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Step 3.5.1

Simplify $16384 \frac{y^{2} y^{\frac{1}{3}}}{16384}$ .

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Step 3.5.1.1

Cancel the common factor of $16384$ .

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Step 3.5.1.1.1

Cancel the common factor.

$16384 \frac{y^{2} y^{\frac{1}{3}}}{16384} = 16384 x$

Step 3.5.1.1.2

Rewrite the expression.

$y^{2} y^{\frac{1}{3}} = 16384 x$

Step 3.5.1.2

Multiply $y^{2}$ by $y^{\frac{1}{3}}$ by adding the exponents.

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Step 3.5.1.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2 + \frac{1}{3}} = 16384 x$

Step 3.5.1.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y^{2 \cdot \frac{3}{3} + \frac{1}{3}} = 16384 x$

Step 3.5.1.2.3

Combine $2$ and $\frac{3}{3}$ .

$y^{\frac{2 \cdot 3}{3} + \frac{1}{3}} = 16384 x$

Step 3.5.1.2.4

Combine the numerators over the common denominator.

$y^{\frac{2 \cdot 3 + 1}{3}} = 16384 x$

Step 3.5.1.2.5

Simplify the numerator.

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Step 3.5.1.2.5.1

Multiply $2$ by $3$ .

$y^{\frac{6 + 1}{3}} = 16384 x$

Step 3.5.1.2.5.2

Add $6$ and $1$ .

$y^{\frac{7}{3}} = 16384 x$

Step 3.6

Raise each side of the equation to the power of $\frac{3}{7}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{7}{3}})}^{\frac{3}{7}} = {(16384 x)}^{\frac{3}{7}}$

Step 3.7

Simplify the exponent.

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Step 3.7.1

Simplify the left side.

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Step 3.7.1.1

Simplify ${(y^{\frac{7}{3}})}^{\frac{3}{7}}$ .

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Step 3.7.1.1.1

Multiply the exponents in ${(y^{\frac{7}{3}})}^{\frac{3}{7}}$ .

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Step 3.7.1.1.1.1

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

$y^{\frac{7}{3} \cdot \frac{3}{7}} = {(16384 x)}^{\frac{3}{7}}$

Step 3.7.1.1.1.2

Cancel the common factor of $7$ .

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Step 3.7.1.1.1.2.1

Cancel the common factor.

$y^{\frac{7}{3} \cdot \frac{3}{7}} = {(16384 x)}^{\frac{3}{7}}$

Step 3.7.1.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{3} \cdot 3} = {(16384 x)}^{\frac{3}{7}}$

Step 3.7.1.1.1.3

Cancel the common factor of $3$ .

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Step 3.7.1.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(16384 x)}^{\frac{3}{7}}$

Step 3.7.1.1.1.3.2

Rewrite the expression.

$y^{1} = {(16384 x)}^{\frac{3}{7}}$

Step 3.7.1.1.2

Simplify.

$y = {(16384 x)}^{\frac{3}{7}}$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify ${(16384 x)}^{\frac{3}{7}}$ .

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Step 3.7.2.1.1

Simplify the expression.

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Step 3.7.2.1.1.1

Apply the product rule to $16384 x$ .

$y = 16384^{\frac{3}{7}} x^{\frac{3}{7}}$

Step 3.7.2.1.1.2

Rewrite $16384$ as $4^{7}$ .

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

Step 3.7.2.1.1.3

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

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

Step 3.7.2.1.2

Cancel the common factor of $7$ .

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Step 3.7.2.1.2.1

Cancel the common factor.

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

Step 3.7.2.1.2.2

Rewrite the expression.

$y = 4^{3} x^{\frac{3}{7}}$

Step 3.7.2.1.3

Raise $4$ to the power of $3$ .

$y = 64 x^{\frac{3}{7}}$

Step 4

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

$f^{- 1} (x) = 64 x^{\frac{3}{7}}$

Step 5

Verify if $f^{- 1} (x) = 64 x^{\frac{3}{7}}$ 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 $f^{- 1}$ .

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 {({(\frac{\sqrt[3]{x}}{4})}^{7})}^{\frac{3}{7}}$

Step 5.2.3

Apply basic rules of exponents.

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Step 5.2.3.1

Multiply the exponents in ${({(\frac{\sqrt[3]{x}}{4})}^{7})}^{\frac{3}{7}}$ .

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Step 5.2.3.1.1

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

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 {(\frac{\sqrt[3]{x}}{4})}^{7 (\frac{3}{7})}$

Step 5.2.3.1.2

Cancel the common factor of $7$ .

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Step 5.2.3.1.2.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 {(\frac{\sqrt[3]{x}}{4})}^{7 (\frac{3}{7})}$

Step 5.2.3.1.2.2

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 {(\frac{\sqrt[3]{x}}{4})}^{3}$

Step 5.2.3.2

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

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 (\frac{{\sqrt[3]{x}}^{3}}{4^{3}})$

Step 5.2.4

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

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Step 5.2.4.1

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

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 (\frac{{(x^{\frac{1}{3}})}^{3}}{4^{3}})$

Step 5.2.4.2

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

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 (\frac{x^{\frac{1}{3} \cdot 3}}{4^{3}})$

Step 5.2.4.3

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

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 (\frac{x^{\frac{3}{3}}}{4^{3}})$

Step 5.2.4.4

Cancel the common factor of $3$ .

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Step 5.2.4.4.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 (\frac{x^{\frac{3}{3}}}{4^{3}})$

Step 5.2.4.4.2

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 (\frac{x}{4^{3}})$

Step 5.2.4.5

Simplify.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 (\frac{x}{4^{3}})$

Step 5.2.5

Raise $4$ to the power of $3$ .

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 (\frac{x}{64})$

Step 5.2.6

Cancel the common factor of $64$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[3]{x}}{4})}^{7}) = 64 (\frac{x}{64})$

Step 5.2.6.2

Rewrite the expression.

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

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 $f (64 x^{\frac{3}{7}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (64 x^{\frac{3}{7}}) = {(\frac{\sqrt[3]{64 x^{\frac{3}{7}}}}{4})}^{7}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Rewrite $64 x^{\frac{3}{7}}$ as ${(4 x^{\frac{1}{7}})}^{3}$ .

$f (64 x^{\frac{3}{7}}) = {(\frac{\sqrt[3]{{(4 x^{\frac{1}{7}})}^{3}}}{4})}^{7}$

Step 5.3.3.2

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

$f (64 x^{\frac{3}{7}}) = {(\frac{4 x^{\frac{1}{7}}}{4})}^{7}$

Step 5.3.4

Reduce the expression by cancelling the common factors.

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Step 5.3.4.1

Cancel the common factor.

$f (64 x^{\frac{3}{7}}) = {(\frac{4 x^{\frac{1}{7}}}{4})}^{7}$

Step 5.3.4.2

Simplify the expression.

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Step 5.3.4.2.1

Divide $x^{\frac{1}{7}}$ by $1$ .

$f (64 x^{\frac{3}{7}}) = {(x^{\frac{1}{7}})}^{7}$

Step 5.3.4.2.2

Multiply the exponents in ${(x^{\frac{1}{7}})}^{7}$ .

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Step 5.3.4.2.2.1

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

$f (64 x^{\frac{3}{7}}) = x^{\frac{1}{7} \cdot 7}$

Step 5.3.4.2.2.2

Cancel the common factor of $7$ .

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Step 5.3.4.2.2.2.1

Cancel the common factor.

$f (64 x^{\frac{3}{7}}) = x^{\frac{1}{7} \cdot 7}$

Step 5.3.4.2.2.2.2

Rewrite the expression.

$f (64 x^{\frac{3}{7}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 64 x^{\frac{3}{7}}$ is the inverse of $f (x) = {(\frac{\sqrt[3]{x}}{4})}^{7}$ .

$f^{- 1} (x) = 64 x^{\frac{3}{7}}$