Precalculus Examples

$y = {(x - 2)}^{3} + 8$

Step 1

Interchange the variables.

$x = {(y - 2)}^{3} + 8$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as ${(y - 2)}^{3} + 8 = x$ .

${(y - 2)}^{3} + 8 = x$

Step 2.2

Subtract $8$ from both sides of the equation.

${(y - 2)}^{3} = x - 8$

Step 2.3

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

$y - 2 = \sqrt[3]{x - 8}$

Step 2.4

Add $2$ to both sides of the equation.

$y = \sqrt[3]{x - 8} + 2$

Step 3

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

$f^{- 1} (x) = \sqrt[3]{x - 8} + 2$

Step 4

Verify if $f^{- 1} (x) = \sqrt[3]{x - 8} + 2$ is the inverse of $f (x) = {(x - 2)}^{3} + 8$ .

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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 $f^{- 1} ({(x - 2)}^{3} + 8)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(x - 2)}^{3} + 8) = \sqrt[3]{({(x - 2)}^{3} + 8) - 8} + 2$

Step 4.2.3

Combine the opposite terms in $\sqrt[3]{({(x - 2)}^{3} + 8) - 8} + 2$ .

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Step 4.2.3.1

Subtract $8$ from $8$ .

$f^{- 1} ({(x - 2)}^{3} + 8) = \sqrt[3]{{(x - 2)}^{3} + 0} + 2$

Step 4.2.3.2

Add ${(x - 2)}^{3}$ and $0$ .

$f^{- 1} ({(x - 2)}^{3} + 8) = \sqrt[3]{{(x - 2)}^{3}} + 2$

Step 4.2.4

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

$f^{- 1} ({(x - 2)}^{3} + 8) = x - 2 + 2$

Step 4.2.5

Combine the opposite terms in $x - 2 + 2$ .

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Step 4.2.5.1

Add $- 2$ and $2$ .

$f^{- 1} ({(x - 2)}^{3} + 8) = x + 0$

Step 4.2.5.2

Add $x$ and $0$ .

$f^{- 1} ({(x - 2)}^{3} + 8) = x$

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 $f (\sqrt[3]{x - 8} + 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[3]{x - 8} + 2) = {((\sqrt[3]{x - 8} + 2) - 2)}^{3} + 8$

Step 4.3.3

Combine the opposite terms in ${((\sqrt[3]{x - 8} + 2) - 2)}^{3} + 8$ .

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Step 4.3.3.1

Subtract $2$ from $2$ .

$f (\sqrt[3]{x - 8} + 2) = {(\sqrt[3]{x - 8} + 0)}^{3} + 8$

Step 4.3.3.2

Add $\sqrt[3]{x - 8}$ and $0$ .

$f (\sqrt[3]{x - 8} + 2) = {\sqrt[3]{x - 8}}^{3} + 8$

Step 4.3.4

Rewrite ${\sqrt[3]{x - 8}}^{3}$ as $x - 8$ .

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Step 4.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 8}$ as ${(x - 8)}^{\frac{1}{3}}$ .

$f (\sqrt[3]{x - 8} + 2) = {({(x - 8)}^{\frac{1}{3}})}^{3} + 8$

Step 4.3.4.2

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

$f (\sqrt[3]{x - 8} + 2) = {(x - 8)}^{\frac{1}{3} \cdot 3} + 8$

Step 4.3.4.3

Combine $\frac{1}{3}$ and $3$ .

$f (\sqrt[3]{x - 8} + 2) = {(x - 8)}^{\frac{3}{3}} + 8$

Step 4.3.4.4

Cancel the common factor of $3$ .

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Step 4.3.4.4.1

Cancel the common factor.

$f (\sqrt[3]{x - 8} + 2) = {(x - 8)}^{\frac{3}{3}} + 8$

Step 4.3.4.4.2

Rewrite the expression.

$f (\sqrt[3]{x - 8} + 2) = (x - 8) + 8$

Step 4.3.4.5

Simplify.

$f (\sqrt[3]{x - 8} + 2) = x - 8 + 8$

Step 4.3.5

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

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Step 4.3.5.1

Add $- 8$ and $8$ .

$f (\sqrt[3]{x - 8} + 2) = x + 0$

Step 4.3.5.2

Add $x$ and $0$ .

$f (\sqrt[3]{x - 8} + 2) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[3]{x - 8} + 2$ is the inverse of $f (x) = {(x - 2)}^{3} + 8$ .

$f^{- 1} (x) = \sqrt[3]{x - 8} + 2$