Find the Inverse f(x)=(x1/4)/5

$f (x) = \frac{x \frac{1}{4}}{5}$

Write $f (x) = \frac{x \frac{1}{4}}{5}$ as an equation.

$y = \frac{x \frac{1}{4}}{5}$

Interchange the variables.

$x = \frac{y (\frac{1}{4})}{5}$

Solve for $y$ .

Rewrite the equation as $\frac{y (\frac{1}{4})}{5} = x$ .

$\frac{y (\frac{1}{4})}{5} = x$

Multiply both sides of the equation by $\frac{5}{\frac{1}{4}}$ .

$\frac{5}{\frac{1}{4}} \cdot \frac{y \frac{1}{4}}{5} = \frac{5}{\frac{1}{4}} x$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{5}{\frac{1}{4}} \cdot \frac{y \frac{1}{4}}{5}$ .

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5}{\frac{1}{4}} \cdot \frac{y \frac{1}{4}}{5} = \frac{5}{\frac{1}{4}} x$

Rewrite the expression.

$\frac{1}{\frac{1}{4}} (y \frac{1}{4}) = \frac{5}{\frac{1}{4}} x$

Cancel the common factor of $\frac{1}{4}$ .

Factor $\frac{1}{4}$ out of $y \frac{1}{4}$ .

$\frac{1}{\frac{1}{4}} (\frac{1}{4} y) = \frac{5}{\frac{1}{4}} x$

Cancel the common factor.

$\frac{1}{\frac{1}{4}} (\frac{1}{4} y) = \frac{5}{\frac{1}{4}} x$

Rewrite the expression.

$y = \frac{5}{\frac{1}{4}} x$

Simplify the right side.

Simplify $\frac{5}{\frac{1}{4}} x$ .

Multiply the numerator by the reciprocal of the denominator.

$y = 5 \cdot 4 x$

Multiply $5$ by $4$ .

$y = 20 x$

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

$f^{- 1} (x) = 20 x$

Verify if $f^{- 1} (x) = 20 x$ is the inverse of $f (x) = \frac{x (\frac{1}{4})}{5}$ .

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 $f^{- 1} (\frac{x (\frac{1}{4})}{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x (\frac{1}{4})}{5}) = 20 (\frac{x (\frac{1}{4})}{5})$

Combine $x$ and $\frac{1}{4}$ .

$f^{- 1} (\frac{x (\frac{1}{4})}{5}) = 20 (\frac{\frac{x}{4}}{5})$

Cancel the common factor of $5$ .

Factor $5$ out of $20$ .

$f^{- 1} (\frac{x (\frac{1}{4})}{5}) = 5 (4) (\frac{\frac{x}{4}}{5})$

Cancel the common factor.

$f^{- 1} (\frac{x (\frac{1}{4})}{5}) = 5 \cdot (4 (\frac{\frac{x}{4}}{5}))$

Rewrite the expression.

$f^{- 1} (\frac{x (\frac{1}{4})}{5}) = 4 (\frac{x}{4})$

Combine $4$ and $\frac{x}{4}$ .

$f^{- 1} (\frac{x (\frac{1}{4})}{5}) = \frac{4 x}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$f^{- 1} (\frac{x (\frac{1}{4})}{5}) = \frac{4 x}{4}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{x (\frac{1}{4})}{5}) = x$

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

Set up the composite result function.

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

Evaluate $f (20 x)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (20 x) = \frac{(20 x) (\frac{1}{4})}{5}$

Cancel the common factor of $20$ and $5$ .

Factor $5$ out of $(20 x) (\frac{1}{4})$ .

$f (20 x) = \frac{5 (4 x (\frac{1}{4}))}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$f (20 x) = \frac{5 (4 x (\frac{1}{4}))}{5 (1)}$

Cancel the common factor.

$f (20 x) = \frac{5 (4 x (\frac{1}{4}))}{5 \cdot 1}$

Rewrite the expression.

$f (20 x) = \frac{4 x (\frac{1}{4})}{1}$

Divide $4 x \frac{1}{4}$ by $1$ .

$f (20 x) = 4 x (\frac{1}{4})$

Cancel the common factor of $4$ .

Factor $4$ out of $4 x$ .

$f (20 x) = 4 (x) (\frac{1}{4})$

Cancel the common factor.

$f (20 x) = 4 x (\frac{1}{4})$

Rewrite the expression.

$f (20 x) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 20 x$ is the inverse of $f (x) = \frac{x (\frac{1}{4})}{5}$ .

$f^{- 1} (x) = 20 x$

Find the Inverse f(x)=(x1/4)/5

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