Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

Apply the product rule to $\frac{\sqrt[7]{y}}{7}$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

$x = \frac{\sqrt[7]{y^{3}}}{343}$

Step 3.2

Rewrite the equation as $\frac{\sqrt[7]{y^{3}}}{343} = x$ .

$\frac{\sqrt[7]{y^{3}}}{343} = x$

Step 3.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{y^{3}}$ as $y^{\frac{3}{7}}$ .

$\frac{y^{\frac{3}{7}}}{343} = x$

Step 3.4

Multiply both sides of the equation by $343$ .

$343 \frac{y^{\frac{3}{7}}}{343} = 343 x$

Step 3.5

Simplify the left side.

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Step 3.5.1

Cancel the common factor of $343$ .

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Step 3.5.1.1

Cancel the common factor.

$343 \frac{y^{\frac{3}{7}}}{343} = 343 x$

Step 3.5.1.2

Rewrite the expression.

$y^{\frac{3}{7}} = 343 x$

Step 3.6

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

${(y^{\frac{3}{7}})}^{\frac{7}{3}} = {(343 x)}^{\frac{7}{3}}$

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

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Step 3.7.1.1.1

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

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Step 3.7.1.1.1.1

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

$y^{\frac{3}{7} \cdot \frac{7}{3}} = {(343 x)}^{\frac{7}{3}}$

Step 3.7.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.7.1.1.1.2.1

Cancel the common factor.

$y^{\frac{3}{7} \cdot \frac{7}{3}} = {(343 x)}^{\frac{7}{3}}$

Step 3.7.1.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{7} \cdot 7} = {(343 x)}^{\frac{7}{3}}$

Step 3.7.1.1.1.3

Cancel the common factor of $7$ .

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Step 3.7.1.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{7} \cdot 7} = {(343 x)}^{\frac{7}{3}}$

Step 3.7.1.1.1.3.2

Rewrite the expression.

$y^{1} = {(343 x)}^{\frac{7}{3}}$

Step 3.7.1.1.2

Simplify.

$y = {(343 x)}^{\frac{7}{3}}$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify ${(343 x)}^{\frac{7}{3}}$ .

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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 $343 x$ .

$y = 343^{\frac{7}{3}} x^{\frac{7}{3}}$

Step 3.7.2.1.1.2

Rewrite $343$ as $7^{3}$ .

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

Step 3.7.2.1.1.3

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

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

Step 3.7.2.1.2

Cancel the common factor of $3$ .

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Step 3.7.2.1.2.1

Cancel the common factor.

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

Step 3.7.2.1.2.2

Rewrite the expression.

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

Step 3.7.2.1.3

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

$y = 823543 x^{\frac{7}{3}}$

Step 4

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

$f^{- 1} (x) = 823543 x^{\frac{7}{3}}$

Step 5

Verify if $f^{- 1} (x) = 823543 x^{\frac{7}{3}}$ is the inverse of $f (x) = {(\frac{\sqrt[7]{x}}{7})}^{3}$ .

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 {({(\frac{\sqrt[7]{x}}{7})}^{3})}^{\frac{7}{3}}$

Step 5.2.3

Apply basic rules of exponents.

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Step 5.2.3.1

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

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Step 5.2.3.1.1

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 {(\frac{\sqrt[7]{x}}{7})}^{3 (\frac{7}{3})}$

Step 5.2.3.1.2

Cancel the common factor of $3$ .

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Step 5.2.3.1.2.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 {(\frac{\sqrt[7]{x}}{7})}^{3 (\frac{7}{3})}$

Step 5.2.3.1.2.2

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 {(\frac{\sqrt[7]{x}}{7})}^{7}$

Step 5.2.3.2

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 (\frac{{\sqrt[7]{x}}^{7}}{7^{7}})$

Step 5.2.4

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

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Step 5.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x}$ as $x^{\frac{1}{7}}$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 (\frac{{(x^{\frac{1}{7}})}^{7}}{7^{7}})$

Step 5.2.4.2

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 (\frac{x^{\frac{1}{7} \cdot 7}}{7^{7}})$

Step 5.2.4.3

Combine $\frac{1}{7}$ and $7$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 (\frac{x^{\frac{7}{7}}}{7^{7}})$

Step 5.2.4.4

Cancel the common factor of $7$ .

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Step 5.2.4.4.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 (\frac{x^{\frac{7}{7}}}{7^{7}})$

Step 5.2.4.4.2

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 (\frac{x}{7^{7}})$

Step 5.2.4.5

Simplify.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 (\frac{x}{7^{7}})$

Step 5.2.5

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 (\frac{x}{823543})$

Step 5.2.6

Cancel the common factor of $823543$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{7})}^{3}) = 823543 (\frac{x}{823543})$

Step 5.2.6.2

Rewrite the expression.

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

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 $f (823543 x^{\frac{7}{3}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (823543 x^{\frac{7}{3}}) = {(\frac{\sqrt[7]{823543 x^{\frac{7}{3}}}}{7})}^{3}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Rewrite $823543 x^{\frac{7}{3}}$ as ${(7 x^{\frac{1}{3}})}^{7}$ .

$f (823543 x^{\frac{7}{3}}) = {(\frac{\sqrt[7]{{(7 x^{\frac{1}{3}})}^{7}}}{7})}^{3}$

Step 5.3.3.2

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

$f (823543 x^{\frac{7}{3}}) = {(\frac{7 x^{\frac{1}{3}}}{7})}^{3}$

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.

$f (823543 x^{\frac{7}{3}}) = {(\frac{7 x^{\frac{1}{3}}}{7})}^{3}$

Step 5.3.4.2

Simplify the expression.

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Step 5.3.4.2.1

Divide $x^{\frac{1}{3}}$ by $1$ .

$f (823543 x^{\frac{7}{3}}) = {(x^{\frac{1}{3}})}^{3}$

Step 5.3.4.2.2

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 5.3.4.2.2.1

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

$f (823543 x^{\frac{7}{3}}) = x^{\frac{1}{3} \cdot 3}$

Step 5.3.4.2.2.2

Cancel the common factor of $3$ .

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Step 5.3.4.2.2.2.1

Cancel the common factor.

$f (823543 x^{\frac{7}{3}}) = x^{\frac{1}{3} \cdot 3}$

Step 5.3.4.2.2.2.2

Rewrite the expression.

$f (823543 x^{\frac{7}{3}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 823543 x^{\frac{7}{3}}$ is the inverse of $f (x) = {(\frac{\sqrt[7]{x}}{7})}^{3}$ .

$f^{- 1} (x) = 823543 x^{\frac{7}{3}}$