Precalculus Examples

$f (x) = {(x - 5)}^{3} + 6$

Step 1

Write $f (x) = {(x - 5)}^{3} + 6$ as an equation.

$y = {(x - 5)}^{3} + 6$

Step 2

Interchange the variables.

$x = {(y - 5)}^{3} + 6$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as ${(y - 5)}^{3} + 6 = x$ .

${(y - 5)}^{3} + 6 = x$

Step 3.2

Subtract $6$ from both sides of the equation.

${(y - 5)}^{3} = x - 6$

Step 3.3

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

$y - 5 = \sqrt[3]{x - 6}$

Step 3.4

Add $5$ to both sides of the equation.

$y = \sqrt[3]{x - 6} + 5$

Step 4

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

$f^{- 1} (x) = \sqrt[3]{x - 6} + 5$

Step 5

Verify if $f^{- 1} (x) = \sqrt[3]{x - 6} + 5$ is the inverse of $f (x) = {(x - 5)}^{3} + 6$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} ({(x - 5)}^{3} + 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(x - 5)}^{3} + 6) = \sqrt[3]{({(x - 5)}^{3} + 6) - 6} + 5$

Step 5.2.3

Combine the opposite terms in $\sqrt[3]{({(x - 5)}^{3} + 6) - 6} + 5$ .

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Step 5.2.3.1

Subtract $6$ from $6$ .

$f^{- 1} ({(x - 5)}^{3} + 6) = \sqrt[3]{{(x - 5)}^{3} + 0} + 5$

Step 5.2.3.2

Add ${(x - 5)}^{3}$ and $0$ .

$f^{- 1} ({(x - 5)}^{3} + 6) = \sqrt[3]{{(x - 5)}^{3}} + 5$

Step 5.2.4

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

$f^{- 1} ({(x - 5)}^{3} + 6) = x - 5 + 5$

Step 5.2.5

Combine the opposite terms in $x - 5 + 5$ .

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Step 5.2.5.1

Add $- 5$ and $5$ .

$f^{- 1} ({(x - 5)}^{3} + 6) = x + 0$

Step 5.2.5.2

Add $x$ and $0$ .

$f^{- 1} ({(x - 5)}^{3} + 6) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\sqrt[3]{x - 6} + 5)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[3]{x - 6} + 5) = {((\sqrt[3]{x - 6} + 5) - 5)}^{3} + 6$

Step 5.3.3

Combine the opposite terms in ${((\sqrt[3]{x - 6} + 5) - 5)}^{3} + 6$ .

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Step 5.3.3.1

Subtract $5$ from $5$ .

$f (\sqrt[3]{x - 6} + 5) = {(\sqrt[3]{x - 6} + 0)}^{3} + 6$

Step 5.3.3.2

Add $\sqrt[3]{x - 6}$ and $0$ .

$f (\sqrt[3]{x - 6} + 5) = {\sqrt[3]{x - 6}}^{3} + 6$

Step 5.3.4

Rewrite ${\sqrt[3]{x - 6}}^{3}$ as $x - 6$ .

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Step 5.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 6}$ as ${(x - 6)}^{\frac{1}{3}}$ .

$f (\sqrt[3]{x - 6} + 5) = {({(x - 6)}^{\frac{1}{3}})}^{3} + 6$

Step 5.3.4.2

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

$f (\sqrt[3]{x - 6} + 5) = {(x - 6)}^{\frac{1}{3} \cdot 3} + 6$

Step 5.3.4.3

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

$f (\sqrt[3]{x - 6} + 5) = {(x - 6)}^{\frac{3}{3}} + 6$

Step 5.3.4.4

Cancel the common factor of $3$ .

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Step 5.3.4.4.1

Cancel the common factor.

$f (\sqrt[3]{x - 6} + 5) = {(x - 6)}^{\frac{3}{3}} + 6$

Step 5.3.4.4.2

Rewrite the expression.

$f (\sqrt[3]{x - 6} + 5) = (x - 6) + 6$

Step 5.3.4.5

Simplify.

$f (\sqrt[3]{x - 6} + 5) = x - 6 + 6$

Step 5.3.5

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

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Step 5.3.5.1

Add $- 6$ and $6$ .

$f (\sqrt[3]{x - 6} + 5) = x + 0$

Step 5.3.5.2

Add $x$ and $0$ .

$f (\sqrt[3]{x - 6} + 5) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[3]{x - 6} + 5$ is the inverse of $f (x) = {(x - 5)}^{3} + 6$ .

$f^{- 1} (x) = \sqrt[3]{x - 6} + 5$