Precalculus Examples

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

Step 1

Write $f (x) = \frac{x^{\frac{1}{3}}}{3}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y^{\frac{1}{3}}}{3} = x$ .

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

Step 3.2

Multiply both sides of the equation by $3$ .

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

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $3$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.4

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Step 3.5

Simplify the exponent.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

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

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Step 3.5.1.1.1

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

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Step 3.5.1.1.1.1

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

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

Step 3.5.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.5.1.1.1.2.1

Cancel the common factor.

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

Step 3.5.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(3 x)}^{3}$

Step 3.5.1.1.2

Simplify.

$y = {(3 x)}^{3}$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

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

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Step 3.5.2.1.1

Apply the product rule to $3 x$ .

$y = 3^{3} x^{3}$

Step 3.5.2.1.2

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

$y = 27 x^{3}$

Step 4

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

$f^{- 1} (x) = 27 x^{3}$

Step 5

Verify if $f^{- 1} (x) = 27 x^{3}$ is the inverse of $f (x) = \frac{x^{\frac{1}{3}}}{3}$ .

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

$f^{- 1} (\frac{x^{\frac{1}{3}}}{3}) = 27 {(\frac{x^{\frac{1}{3}}}{3})}^{3}$

Step 5.2.3

Apply the product rule to $\frac{x^{\frac{1}{3}}}{3}$ .

$f^{- 1} (\frac{x^{\frac{1}{3}}}{3}) = 27 (\frac{{(x^{\frac{1}{3}})}^{3}}{3^{3}})$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

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

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Step 5.2.4.1.1

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

$f^{- 1} (\frac{x^{\frac{1}{3}}}{3}) = 27 (\frac{x^{\frac{1}{3} \cdot 3}}{3^{3}})$

Step 5.2.4.1.2

Cancel the common factor of $3$ .

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Step 5.2.4.1.2.1

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{3}) = 27 (\frac{x^{\frac{1}{3} \cdot 3}}{3^{3}})$

Step 5.2.4.1.2.2

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{3}) = 27 (\frac{x}{3^{3}})$

Step 5.2.4.2

Simplify.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{3}) = 27 (\frac{x}{3^{3}})$

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

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

$f^{- 1} (\frac{x^{\frac{1}{3}}}{3}) = 27 (\frac{x}{27})$

Step 5.2.5.2

Cancel the common factor of $27$ .

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Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{3}}}{3}) = 27 (\frac{x}{27})$

Step 5.2.5.2.2

Rewrite the expression.

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

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 $f (27 x^{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (27 x^{3}) = \frac{{(27 x^{3})}^{\frac{1}{3}}}{3}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Apply the product rule to $27 x^{3}$ .

$f (27 x^{3}) = \frac{27^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}{3}$

Step 5.3.3.2

Rewrite $27$ as $3^{3}$ .

$f (27 x^{3}) = \frac{{(3^{3})}^{\frac{1}{3}} {(x^{3})}^{\frac{1}{3}}}{3}$

Step 5.3.3.3

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

$f (27 x^{3}) = \frac{3^{3 (\frac{1}{3})} {(x^{3})}^{\frac{1}{3}}}{3}$

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.

$f (27 x^{3}) = \frac{3^{3 (\frac{1}{3})} {(x^{3})}^{\frac{1}{3}}}{3}$

Step 5.3.3.4.2

Rewrite the expression.

$f (27 x^{3}) = \frac{3 {(x^{3})}^{\frac{1}{3}}}{3}$

Step 5.3.3.5

Evaluate the exponent.

$f (27 x^{3}) = \frac{3 {(x^{3})}^{\frac{1}{3}}}{3}$

Step 5.3.3.6

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

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Step 5.3.3.6.1

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

$f (27 x^{3}) = \frac{3 x^{3 (\frac{1}{3})}}{3}$

Step 5.3.3.6.2

Cancel the common factor of $3$ .

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Step 5.3.3.6.2.1

Cancel the common factor.

$f (27 x^{3}) = \frac{3 x^{3 (\frac{1}{3})}}{3}$

Step 5.3.3.6.2.2

Rewrite the expression.

$f (27 x^{3}) = \frac{3 x}{3}$

Step 5.3.3.7

Simplify.

$f (27 x^{3}) = \frac{3 x}{3}$

Step 5.3.4

Cancel the common factor of $3$ .

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Step 5.3.4.1

Cancel the common factor.

$f (27 x^{3}) = \frac{3 x}{3}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (27 x^{3}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 27 x^{3}$ is the inverse of $f (x) = \frac{x^{\frac{1}{3}}}{3}$ .

$f^{- 1} (x) = 27 x^{3}$