Find the Inverse F(x)=(x+8)^5

$F (x) = {(x + 8)}^{5}$

Step 1

Write $F (x) = {(x + 8)}^{5}$ as an equation.

$y = {(x + 8)}^{5}$

Step 2

Interchange the variables.

$x = {(y + 8)}^{5}$

Step 3

Solve for $y$ .

Rewrite the equation as ${(y + 8)}^{5} = x$ .

${(y + 8)}^{5} = x$

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

$y + 8 = \sqrt[5]{x}$

Subtract $8$ from both sides of the equation.

$y = \sqrt[5]{x} - 8$

Step 4

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

$F^{- 1} (x) = \sqrt[5]{x} - 8$

Step 5

Verify if $F^{- 1} (x) = \sqrt[5]{x} - 8$ is the inverse of $F (x) = {(x + 8)}^{5}$ .

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 $F^{- 1} ({(x + 8)}^{5})$ by substituting in the value of $F$ into $F^{- 1}$ .

$F^{- 1} ({(x + 8)}^{5}) = \sqrt[5]{{(x + 8)}^{5}} - 8$

Remove parentheses.

$F^{- 1} ({(x + 8)}^{5}) = \sqrt[5]{{(x + 8)}^{5}} - 8$

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

$F^{- 1} ({(x + 8)}^{5}) = x + 8 - 8$

Combine the opposite terms in $x + 8 - 8$ .

Subtract $8$ from $8$ .

$F^{- 1} ({(x + 8)}^{5}) = x + 0$

Add $x$ and $0$ .

$F^{- 1} ({(x + 8)}^{5}) = x$

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

Set up the composite result function.

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

Evaluate $F (\sqrt[5]{x} - 8)$ by substituting in the value of $F^{- 1}$ into $F$ .

$F (\sqrt[5]{x} - 8) = {((\sqrt[5]{x} - 8) + 8)}^{5}$

Combine the opposite terms in $(\sqrt[5]{x} - 8) + 8$ .

Add $- 8$ and $8$ .

$F (\sqrt[5]{x} - 8) = {(\sqrt[5]{x} + 0)}^{5}$

Add $\sqrt[5]{x}$ and $0$ .

$F (\sqrt[5]{x} - 8) = {\sqrt[5]{x}}^{5}$

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

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

$F (\sqrt[5]{x} - 8) = {(x^{\frac{1}{5}})}^{5}$

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

$F (\sqrt[5]{x} - 8) = x^{\frac{1}{5} \cdot 5}$

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

$F (\sqrt[5]{x} - 8) = x^{\frac{5}{5}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$F (\sqrt[5]{x} - 8) = x^{\frac{5}{5}}$

Rewrite the expression.

$F (\sqrt[5]{x} - 8) = x$

Simplify.

$F (\sqrt[5]{x} - 8) = x$

Since $F^{- 1} (F (x)) = x$ and $F (F^{- 1} (x)) = x$ , then $F^{- 1} (x) = \sqrt[5]{x} - 8$ is the inverse of $F (x) = {(x + 8)}^{5}$ .

$F^{- 1} (x) = \sqrt[5]{x} - 8$

Find the Inverse F(x)=(x+8)^5

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