Algebra Examples

$y = {(\frac{1}{4} x + 6)}^{3}$

Step 1

Interchange the variables.

$x = {(\frac{1}{4} y + 6)}^{3}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as ${(\frac{1}{4} y + 6)}^{3} = x$ .

${(\frac{1}{4} y + 6)}^{3} = x$

Step 2.2

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

$\frac{1}{4} y + 6 = \sqrt[3]{x}$

Step 2.3

Combine $\frac{1}{4}$ and $y$ .

$\frac{y}{4} + 6 = \sqrt[3]{x}$

Step 2.4

Subtract $6$ from both sides of the equation.

$\frac{y}{4} = \sqrt[3]{x} - 6$

Step 2.5

Multiply both sides of the equation by $4$ .

$4 \frac{y}{4} = 4 (\sqrt[3]{x} - 6)$

Step 2.6

Simplify both sides of the equation.

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Step 2.6.1

Simplify the left side.

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Step 2.6.1.1

Cancel the common factor of $4$ .

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Step 2.6.1.1.1

Cancel the common factor.

$4 \frac{y}{4} = 4 (\sqrt[3]{x} - 6)$

Step 2.6.1.1.2

Rewrite the expression.

$y = 4 (\sqrt[3]{x} - 6)$

Step 2.6.2

Simplify the right side.

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Step 2.6.2.1

Simplify $4 (\sqrt[3]{x} - 6)$ .

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Step 2.6.2.1.1

Apply the distributive property.

$y = 4 \sqrt[3]{x} + 4 \cdot - 6$

Step 2.6.2.1.2

Multiply $4$ by $- 6$ .

$y = 4 \sqrt[3]{x} - 24$

Step 3

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

$f^{- 1} (x) = 4 \sqrt[3]{x} - 24$

Step 4

Verify if $f^{- 1} (x) = 4 \sqrt[3]{x} - 24$ is the inverse of $f (x) = {(\frac{1}{4} x + 6)}^{3}$ .

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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 $f^{- 1} ({(\frac{1}{4} x + 6)}^{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(\frac{1}{4} x + 6)}^{3}) = 4 \sqrt[3]{{(\frac{1}{4} x + 6)}^{3}} - 24$

Step 4.2.3

Remove parentheses.

$f^{- 1} ({(\frac{1}{4} x + 6)}^{3}) = 4 \sqrt[3]{{(\frac{1}{4} x + 6)}^{3}} - 24$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

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

$f^{- 1} ({(\frac{1}{4} x + 6)}^{3}) = 4 (\frac{1}{4} x + 6) - 24$

Step 4.2.4.2

Combine $\frac{1}{4}$ and $x$ .

$f^{- 1} ({(\frac{1}{4} x + 6)}^{3}) = 4 (\frac{x}{4} + 6) - 24$

Step 4.2.4.3

Apply the distributive property.

$f^{- 1} ({(\frac{1}{4} x + 6)}^{3}) = 4 (\frac{x}{4}) + 4 \cdot 6 - 24$

Step 4.2.4.4

Cancel the common factor of $4$ .

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Step 4.2.4.4.1

Cancel the common factor.

$f^{- 1} ({(\frac{1}{4} x + 6)}^{3}) = 4 (\frac{x}{4}) + 4 \cdot 6 - 24$

Step 4.2.4.4.2

Rewrite the expression.

$f^{- 1} ({(\frac{1}{4} x + 6)}^{3}) = x + 4 \cdot 6 - 24$

Step 4.2.4.5

Multiply $4$ by $6$ .

$f^{- 1} ({(\frac{1}{4} x + 6)}^{3}) = x + 24 - 24$

Step 4.2.5

Combine the opposite terms in $x + 24 - 24$ .

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Step 4.2.5.1

Subtract $24$ from $24$ .

$f^{- 1} ({(\frac{1}{4} x + 6)}^{3}) = x + 0$

Step 4.2.5.2

Add $x$ and $0$ .

$f^{- 1} ({(\frac{1}{4} x + 6)}^{3}) = x$

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 $f (4 \sqrt[3]{x} - 24)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (4 \sqrt[3]{x} - 24) = {(\frac{1}{4} \cdot (4 \sqrt[3]{x} - 24) + 6)}^{3}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (4 \sqrt[3]{x} - 24) = {(\frac{1}{4} \cdot (4 \sqrt[3]{x}) + \frac{1}{4} \cdot - 24 + 6)}^{3}$

Step 4.3.3.2

Cancel the common factor of $4$ .

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Step 4.3.3.2.1

Factor $4$ out of $4 \sqrt[3]{x}$ .

$f (4 \sqrt[3]{x} - 24) = {(\frac{1}{4} \cdot (4 (\sqrt[3]{x})) + \frac{1}{4} \cdot - 24 + 6)}^{3}$

Step 4.3.3.2.2

Cancel the common factor.

$f (4 \sqrt[3]{x} - 24) = {(\frac{1}{4} \cdot (4 \sqrt[3]{x}) + \frac{1}{4} \cdot - 24 + 6)}^{3}$

Step 4.3.3.2.3

Rewrite the expression.

$f (4 \sqrt[3]{x} - 24) = {(\sqrt[3]{x} + \frac{1}{4} \cdot - 24 + 6)}^{3}$

Step 4.3.3.3

Cancel the common factor of $4$ .

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Step 4.3.3.3.1

Factor $4$ out of $- 24$ .

$f (4 \sqrt[3]{x} - 24) = {(\sqrt[3]{x} + \frac{1}{4} \cdot (4 (- 6)) + 6)}^{3}$

Step 4.3.3.3.2

Cancel the common factor.

$f (4 \sqrt[3]{x} - 24) = {(\sqrt[3]{x} + \frac{1}{4} \cdot (4 \cdot - 6) + 6)}^{3}$

Step 4.3.3.3.3

Rewrite the expression.

$f (4 \sqrt[3]{x} - 24) = {(\sqrt[3]{x} - 6 + 6)}^{3}$

Step 4.3.4

Simplify terms.

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Step 4.3.4.1

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

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Step 4.3.4.1.1

Add $- 6$ and $6$ .

$f (4 \sqrt[3]{x} - 24) = {(\sqrt[3]{x} + 0)}^{3}$

Step 4.3.4.1.2

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

$f (4 \sqrt[3]{x} - 24) = {\sqrt[3]{x}}^{3}$

Step 4.3.4.2

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

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Step 4.3.4.2.1

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

$f (4 \sqrt[3]{x} - 24) = {(x^{\frac{1}{3}})}^{3}$

Step 4.3.4.2.2

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

$f (4 \sqrt[3]{x} - 24) = x^{\frac{1}{3} \cdot 3}$

Step 4.3.4.2.3

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

$f (4 \sqrt[3]{x} - 24) = x^{\frac{3}{3}}$

Step 4.3.4.2.4

Cancel the common factor of $3$ .

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Step 4.3.4.2.4.1

Cancel the common factor.

$f (4 \sqrt[3]{x} - 24) = x^{\frac{3}{3}}$

Step 4.3.4.2.4.2

Rewrite the expression.

$f (4 \sqrt[3]{x} - 24) = x$

Step 4.3.4.2.5

Simplify.

$f (4 \sqrt[3]{x} - 24) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 4 \sqrt[3]{x} - 24$ is the inverse of $f (x) = {(\frac{1}{4} x + 6)}^{3}$ .

$f^{- 1} (x) = 4 \sqrt[3]{x} - 24$