Precalculus Examples

$f (x) = 5 x^{\frac{1}{5}}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = 5 x^{\frac{1}{5}}$ as an equation.

$y = 5 x^{\frac{1}{5}}$

Step 2

Interchange the variables.

$x = 5 y^{\frac{1}{5}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $5 y^{\frac{1}{5}} = x$ .

$5 y^{\frac{1}{5}} = x$

Step 3.2

Divide each term in $5 y^{\frac{1}{5}} = x$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 y^{\frac{1}{5}} = x$ by $5$ .

$\frac{5 y^{\frac{1}{5}}}{5} = \frac{x}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor.

$\frac{5 y^{\frac{1}{5}}}{5} = \frac{x}{5}$

Step 3.2.2.2

Divide $y^{\frac{1}{5}}$ by $1$ .

$y^{\frac{1}{5}} = \frac{x}{5}$

Step 3.3

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{5}})}^{5} = {(\frac{x}{5})}^{5}$

Step 3.4

Simplify the exponent.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify ${(y^{\frac{1}{5}})}^{5}$ .

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Step 3.4.1.1.1

Multiply the exponents in ${(y^{\frac{1}{5}})}^{5}$ .

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Step 3.4.1.1.1.1

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

$y^{\frac{1}{5} \cdot 5} = {(\frac{x}{5})}^{5}$

Step 3.4.1.1.1.2

Cancel the common factor of $5$ .

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Step 3.4.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{5} \cdot 5} = {(\frac{x}{5})}^{5}$

Step 3.4.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(\frac{x}{5})}^{5}$

Step 3.4.1.1.2

Simplify.

$y = {(\frac{x}{5})}^{5}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify ${(\frac{x}{5})}^{5}$ .

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Step 3.4.2.1.1

Apply the product rule to $\frac{x}{5}$ .

$y = \frac{x^{5}}{5^{5}}$

Step 3.4.2.1.2

Raise $5$ to the power of $5$ .

$y = \frac{x^{5}}{3125}$

Step 4

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

$f^{- 1} (x) = \frac{x^{5}}{3125}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{5}}{3125}$ is the inverse of $f (x) = 5 x^{\frac{1}{5}}$ .

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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 $f^{- 1} (5 x^{\frac{1}{5}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 x^{\frac{1}{5}}) = \frac{{(5 x^{\frac{1}{5}})}^{5}}{3125}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}}) = \frac{5^{5} {(x^{\frac{1}{5}})}^{5}}{3125}$

Step 5.2.3.2

Raise $5$ to the power of $5$ .

$f^{- 1} (5 x^{\frac{1}{5}}) = \frac{3125 {(x^{\frac{1}{5}})}^{5}}{3125}$

Step 5.2.3.3

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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Step 5.2.3.3.1

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

$f^{- 1} (5 x^{\frac{1}{5}}) = \frac{3125 x^{\frac{1}{5} \cdot 5}}{3125}$

Step 5.2.3.3.2

Cancel the common factor of $5$ .

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Step 5.2.3.3.2.1

Cancel the common factor.

$f^{- 1} (5 x^{\frac{1}{5}}) = \frac{3125 x^{\frac{1}{5} \cdot 5}}{3125}$

Step 5.2.3.3.2.2

Rewrite the expression.

$f^{- 1} (5 x^{\frac{1}{5}}) = \frac{3125 x}{3125}$

Step 5.2.3.4

Simplify.

$f^{- 1} (5 x^{\frac{1}{5}}) = \frac{3125 x}{3125}$

Step 5.2.4

Cancel the common factor of $3125$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (5 x^{\frac{1}{5}}) = \frac{3125 x}{3125}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (5 x^{\frac{1}{5}}) = x$

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 $f (\frac{x^{5}}{3125})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{5}}{3125}) = 5 {(\frac{x^{5}}{3125})}^{\frac{1}{5}}$

Step 5.3.3

Apply the product rule to $\frac{x^{5}}{3125}$ .

$f (\frac{x^{5}}{3125}) = 5 (\frac{{(x^{5})}^{\frac{1}{5}}}{3125^{\frac{1}{5}}})$

Step 5.3.4

Simplify the numerator.

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Step 5.3.4.1

Multiply the exponents in ${(x^{5})}^{\frac{1}{5}}$ .

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Step 5.3.4.1.1

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

$f (\frac{x^{5}}{3125}) = 5 (\frac{x^{5 (\frac{1}{5})}}{3125^{\frac{1}{5}}})$

Step 5.3.4.1.2

Cancel the common factor of $5$ .

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Step 5.3.4.1.2.1

Cancel the common factor.

$f (\frac{x^{5}}{3125}) = 5 (\frac{x^{5 (\frac{1}{5})}}{3125^{\frac{1}{5}}})$

Step 5.3.4.1.2.2

Rewrite the expression.

$f (\frac{x^{5}}{3125}) = 5 (\frac{x}{3125^{\frac{1}{5}}})$

Step 5.3.4.2

Simplify.

$f (\frac{x^{5}}{3125}) = 5 (\frac{x}{3125^{\frac{1}{5}}})$

Step 5.3.5

Simplify the denominator.

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Step 5.3.5.1

Rewrite $3125$ as $5^{5}$ .

$f (\frac{x^{5}}{3125}) = 5 (\frac{x}{{(5^{5})}^{\frac{1}{5}}})$

Step 5.3.5.2

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

$f (\frac{x^{5}}{3125}) = 5 (\frac{x}{5^{5 (\frac{1}{5})}})$

Step 5.3.5.3

Cancel the common factor of $5$ .

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Step 5.3.5.3.1

Cancel the common factor.

$f (\frac{x^{5}}{3125}) = 5 (\frac{x}{5^{5 (\frac{1}{5})}})$

Step 5.3.5.3.2

Rewrite the expression.

$f (\frac{x^{5}}{3125}) = 5 (\frac{x}{5})$

Step 5.3.5.4

Evaluate the exponent.

$f (\frac{x^{5}}{3125}) = 5 (\frac{x}{5})$

Step 5.3.6

Cancel the common factor of $5$ .

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Step 5.3.6.1

Cancel the common factor.

$f (\frac{x^{5}}{3125}) = 5 (\frac{x}{5})$

Step 5.3.6.2

Rewrite the expression.

$f (\frac{x^{5}}{3125}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{5}}{3125}$ is the inverse of $f (x) = 5 x^{\frac{1}{5}}$ .

$f^{- 1} (x) = \frac{x^{5}}{3125}$