Find the Inverse h(x)=x^5+6

$h (x) = x^{5} + 6$

Step 1

Write $h (x) = x^{5} + 6$ as an equation.

$y = x^{5} + 6$

Step 2

Interchange the variables.

$x = y^{5} + 6$

Step 3

Solve for $y$ .

Rewrite the equation as $y^{5} + 6 = x$ .

$y^{5} + 6 = x$

Subtract $6$ from both sides of the equation.

$y^{5} = x - 6$

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

$y = \sqrt[5]{x - 6}$

Step 4

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

$h^{- 1} (x) = \sqrt[5]{x - 6}$

Step 5

Verify if $h^{- 1} (x) = \sqrt[5]{x - 6}$ is the inverse of $h (x) = x^{5} + 6$ .

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

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

Set up the composite result function.

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

Evaluate $h^{- 1} (x^{5} + 6)$ by substituting in the value of $h$ into $h^{- 1}$ .

$h^{- 1} (x^{5} + 6) = \sqrt[5]{(x^{5} + 6) - 6}$

Subtract $6$ from $6$ .

$h^{- 1} (x^{5} + 6) = \sqrt[5]{x^{5} + 0}$

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

$h^{- 1} (x^{5} + 6) = \sqrt[5]{x^{5}}$

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

$h^{- 1} (x^{5} + 6) = x$

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

Set up the composite result function.

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

Evaluate $h (\sqrt[5]{x - 6})$ by substituting in the value of $h^{- 1}$ into $h$ .

$h (\sqrt[5]{x - 6}) = {(\sqrt[5]{x - 6})}^{5} + 6$

Rewrite ${\sqrt[5]{x - 6}}^{5}$ as $x - 6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x - 6}$ as ${(x - 6)}^{\frac{1}{5}}$ .

$h (\sqrt[5]{x - 6}) = {({(x - 6)}^{\frac{1}{5}})}^{5} + 6$

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

$h (\sqrt[5]{x - 6}) = {(x - 6)}^{\frac{1}{5} \cdot 5} + 6$

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

$h (\sqrt[5]{x - 6}) = {(x - 6)}^{\frac{5}{5}} + 6$

Cancel the common factor of $5$ .

Cancel the common factor.

$h (\sqrt[5]{x - 6}) = {(x - 6)}^{\frac{5}{5}} + 6$

Rewrite the expression.

$h (\sqrt[5]{x - 6}) = (x - 6) + 6$

Simplify.

$h (\sqrt[5]{x - 6}) = x - 6 + 6$

Combine the opposite terms in $x - 6 + 6$ .

Add $- 6$ and $6$ .

$h (\sqrt[5]{x - 6}) = x + 0$

Add $x$ and $0$ .

$h (\sqrt[5]{x - 6}) = x$

Since $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ , then $h^{- 1} (x) = \sqrt[5]{x - 6}$ is the inverse of $h (x) = x^{5} + 6$ .

$h^{- 1} (x) = \sqrt[5]{x - 6}$

Find the Inverse h(x)=x^5+6

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