Find the Inverse f(x)=( fifth root of x)/5

$f (x) = \frac{\sqrt[5]{x}}{5}$

Step 1

Write $f (x) = \frac{\sqrt[5]{x}}{5}$ as an equation.

$y = \frac{\sqrt[5]{x}}{5}$

Step 2

Interchange the variables.

$x = \frac{\sqrt[5]{y}}{5}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{\sqrt[5]{y}}{5} = x$ .

$\frac{\sqrt[5]{y}}{5} = x$

Multiply both sides of the equation by $5$ .

$5 \frac{\sqrt[5]{y}}{5} = 5 x$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$5 \frac{\sqrt[5]{y}}{5} = 5 x$

Rewrite the expression.

$\sqrt[5]{y} = 5 x$

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{y}}^{5} = {(5 x)}^{5}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y}$ as $y^{\frac{1}{5}}$ .

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Apply the product rule to $5 x$ .

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

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

$y = 3125 x^{5}$

Step 4

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

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

Step 5

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

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

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

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

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

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

Rewrite ${\sqrt[5]{x}}^{5}$ as $x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

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

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

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

Combine $\frac{1}{5}$ and $5$ .

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

Cancel the common factor of $3125$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

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

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

Simplify the numerator.

Rewrite $3125 x^{5}$ as ${(5 x)}^{5}$ .

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

Find the Inverse f(x)=( fifth root of x)/5

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