Find the Inverse f(x)=(x^5)/7

$f (x) = \frac{x^{5}}{7}$

Write $f (x) = \frac{x^{5}}{7}$ as an equation.

$y = \frac{x^{5}}{7}$

Interchange the variables.

$x = \frac{y^{5}}{7}$

Solve for $y$ .

Rewrite the equation as $\frac{y^{5}}{7} = x$ .

$\frac{y^{5}}{7} = x$

Multiply both sides of the equation by $7$ .

$7 \frac{y^{5}}{7} = 7 x$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$7 \frac{y^{5}}{7} = 7 x$

Rewrite the expression.

$y^{5} = 7 x$

Take the 5th root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[5]{7 x}$

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

$f^{- 1} (x) = \sqrt[5]{7 x}$

Verify if $f^{- 1} (x) = \sqrt[5]{7 x}$ is the inverse of $f (x) = \frac{x^{5}}{7}$ .

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 $f^{- 1} (\frac{x^{5}}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x^{5}}{7}) = \sqrt[5]{7 (\frac{x^{5}}{7})}$

Combine $7$ and $\frac{x^{5}}{7}$ .

$f^{- 1} (\frac{x^{5}}{7}) = \sqrt[5]{\frac{7 x^{5}}{7}}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{7 x^{5}}{7}$ by cancelling the common factors.

Cancel the common factor.

$f^{- 1} (\frac{x^{5}}{7}) = \sqrt[5]{\frac{7 x^{5}}{7}}$

Rewrite the expression.

$f^{- 1} (\frac{x^{5}}{7}) = \sqrt[5]{\frac{x^{5}}{1}}$

Divide $x^{5}$ by $1$ .

$f^{- 1} (\frac{x^{5}}{7}) = \sqrt[5]{x^{5}}$

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

$f^{- 1} (\frac{x^{5}}{7}) = x$

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

Set up the composite result function.

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

Evaluate $f (\sqrt[5]{7 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[5]{7 x}) = \frac{{(\sqrt[5]{7 x})}^{5}}{7}$

Rewrite ${\sqrt[5]{7 x}}^{5}$ as $7 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{7 x}$ as ${(7 x)}^{\frac{1}{5}}$ .

$f (\sqrt[5]{7 x}) = \frac{{({(7 x)}^{\frac{1}{5}})}^{5}}{7}$

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

$f (\sqrt[5]{7 x}) = \frac{{(7 x)}^{\frac{1}{5} \cdot 5}}{7}$

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

$f (\sqrt[5]{7 x}) = \frac{{(7 x)}^{\frac{5}{5}}}{7}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\sqrt[5]{7 x}) = \frac{{(7 x)}^{\frac{5}{5}}}{7}$

Rewrite the expression.

$f (\sqrt[5]{7 x}) = \frac{7 x}{7}$

Simplify.

$f (\sqrt[5]{7 x}) = \frac{7 x}{7}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\sqrt[5]{7 x}) = \frac{7 x}{7}$

Divide $x$ by $1$ .

$f (\sqrt[5]{7 x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[5]{7 x}$ is the inverse of $f (x) = \frac{x^{5}}{7}$ .

$f^{- 1} (x) = \sqrt[5]{7 x}$

Find the Inverse f(x)=(x^5)/7

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