Trigonometry Examples

$x^{3 + 512}$

Step 1

Interchange the variables.

$x = y^{3 + 512}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $y^{3 + 512} = x$ .

$y^{3 + 512} = x$

Step 2.2

Add $3$ and $512$ .

$y^{515} = x$

Step 2.3

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

$y = \sqrt[515]{x}$

Step 3

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

$f^{- 1} (x) = \sqrt[515]{x}$

Step 4

Verify if $f^{- 1} (x) = \sqrt[515]{x}$ is the inverse of $f (x) = x^{3 + 512}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (x^{3 + 512})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (x^{3 + 512}) = \sqrt[515]{x^{3 + 512}}$

Step 4.2.3

Remove parentheses.

$f^{- 1} (x^{3 + 512}) = \sqrt[515]{x^{3 + 512}}$

Step 4.2.4

Add $3$ and $512$ .

$f^{- 1} (x^{3 + 512}) = \sqrt[515]{x^{515}}$

Step 4.2.5

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

$f^{- 1} (x^{3 + 512}) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\sqrt[515]{x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[515]{x}) = {(\sqrt[515]{x})}^{3 + 512}$

Step 4.3.3

Add $3$ and $512$ .

$f (\sqrt[515]{x}) = {\sqrt[515]{x}}^{515}$

Step 4.3.4

Rewrite ${\sqrt[515]{x}}^{515}$ as $x$ .

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Step 4.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[515]{x}$ as $x^{\frac{1}{515}}$ .

$f (\sqrt[515]{x}) = {(x^{\frac{1}{515}})}^{515}$

Step 4.3.4.2

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

$f (\sqrt[515]{x}) = x^{\frac{1}{515} \cdot 515}$

Step 4.3.4.3

Combine $\frac{1}{515}$ and $515$ .

$f (\sqrt[515]{x}) = x^{\frac{515}{515}}$

Step 4.3.4.4

Cancel the common factor of $515$ .

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Step 4.3.4.4.1

Cancel the common factor.

$f (\sqrt[515]{x}) = x^{\frac{515}{515}}$

Step 4.3.4.4.2

Rewrite the expression.

$f (\sqrt[515]{x}) = x$

Step 4.3.4.5

Simplify.

$f (\sqrt[515]{x}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[515]{x}$ is the inverse of $f (x) = x^{3 + 512}$ .

$f^{- 1} (x) = \sqrt[515]{x}$