Find the Inverse f(x) = fifth root of x^7+7

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

Step 1

Write $f (x) = \sqrt[5]{x^{7} + 7}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

Rewrite the equation as $\sqrt[5]{y^{7} + 7} = x$ .

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

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

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

Simplify each side of the equation.

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

${({(y^{7} + 7)}^{\frac{1}{5}})}^{5} = x^{5}$

Simplify the left side.

Simplify ${({(y^{7} + 7)}^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${({(y^{7} + 7)}^{\frac{1}{5}})}^{5}$ .

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

${(y^{7} + 7)}^{\frac{1}{5} \cdot 5} = x^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(y^{7} + 7)}^{\frac{1}{5} \cdot 5} = x^{5}$

Rewrite the expression.

${(y^{7} + 7)}^{1} = x^{5}$

Simplify.

$y^{7} + 7 = x^{5}$

Solve for $y$ .

Subtract $7$ from both sides of the equation.

$y^{7} = x^{5} - 7$

Take the 7th root of both sides of the equation to eliminate the exponent on the left side.

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

Step 4

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

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

Step 5

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

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

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

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

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

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

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

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify by subtracting numbers.

Subtract $7$ from $7$ .

$f^{- 1} (\sqrt[5]{x^{7} + 7}) = \sqrt[7]{x^{7} + 0}$

Add $x^{7}$ and $0$ .

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

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

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

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

Set up the composite result function.

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

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

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

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

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

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

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

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

Combine $\frac{1}{7}$ and $7$ .

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

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify by adding numbers.

Add $- 7$ and $7$ .

$f (\sqrt[7]{x^{5} - 7}) = \sqrt[5]{x^{5} + 0}$

Add $x^{5}$ and $0$ .

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

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

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

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

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

Find the Inverse f(x) = fifth root of x^7+7

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