Precalculus Examples

$f (x) = \sqrt[4]{x} + 4$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \sqrt[4]{x} + 4$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[4]{y} + 4 = x$ .

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

Step 3.2

Subtract $4$ from both sides of the equation.

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

Step 3.3

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

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

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{y}$ as $y^{\frac{1}{4}}$ .

${(y^{\frac{1}{4}})}^{4} = {(x - 4)}^{4}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify ${(y^{\frac{1}{4}})}^{4}$ .

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Step 3.4.2.1.1

Multiply the exponents in ${(y^{\frac{1}{4}})}^{4}$ .

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Step 3.4.2.1.1.1

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

$y^{\frac{1}{4} \cdot 4} = {(x - 4)}^{4}$

Step 3.4.2.1.1.2

Cancel the common factor of $4$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{4} \cdot 4} = {(x - 4)}^{4}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(x - 4)}^{4}$

Step 3.4.2.1.2

Simplify.

$y = {(x - 4)}^{4}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x - 4)}^{4}$ .

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Step 3.4.3.1.1

Use the Binomial Theorem.

$y = x^{4} + 4 x^{3} \cdot - 4 + 6 x^{2} {(- 4)}^{2} + 4 x {(- 4)}^{3} + {(- 4)}^{4}$

Step 3.4.3.1.2

Simplify each term.

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Step 3.4.3.1.2.1

Multiply $- 4$ by $4$ .

$y = x^{4} - 16 x^{3} + 6 x^{2} {(- 4)}^{2} + 4 x {(- 4)}^{3} + {(- 4)}^{4}$

Step 3.4.3.1.2.2

Raise $- 4$ to the power of $2$ .

$y = x^{4} - 16 x^{3} + 6 x^{2} \cdot 16 + 4 x {(- 4)}^{3} + {(- 4)}^{4}$

Step 3.4.3.1.2.3

Multiply $16$ by $6$ .

$y = x^{4} - 16 x^{3} + 96 x^{2} + 4 x {(- 4)}^{3} + {(- 4)}^{4}$

Step 3.4.3.1.2.4

Raise $- 4$ to the power of $3$ .

$y = x^{4} - 16 x^{3} + 96 x^{2} + 4 x \cdot - 64 + {(- 4)}^{4}$

Step 3.4.3.1.2.5

Multiply $- 64$ by $4$ .

$y = x^{4} - 16 x^{3} + 96 x^{2} - 256 x + {(- 4)}^{4}$

Step 3.4.3.1.2.6

Raise $- 4$ to the power of $4$ .

$y = x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256$

Step 4

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

$f^{- 1} (x) = x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256$

Step 5

Verify if $f^{- 1} (x) = x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256$ is the inverse of $f (x) = \sqrt[4]{x} + 4$ .

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

$f^{- 1} (\sqrt[4]{x} + 4) = {(\sqrt[4]{x} + 4)}^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Use the Binomial Theorem.

$f^{- 1} (\sqrt[4]{x} + 4) = {\sqrt[4]{x}}^{4} + 4 {\sqrt[4]{x}}^{3} \cdot 4 + 6 {\sqrt[4]{x}}^{2} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2

Simplify each term.

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Step 5.2.3.2.1

Rewrite ${\sqrt[4]{x}}^{4}$ as $x$ .

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Step 5.2.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = {(x^{\frac{1}{4}})}^{4} + 4 {\sqrt[4]{x}}^{3} \cdot 4 + 6 {\sqrt[4]{x}}^{2} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.1.2

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

$f^{- 1} (\sqrt[4]{x} + 4) = x^{\frac{1}{4} \cdot 4} + 4 {\sqrt[4]{x}}^{3} \cdot 4 + 6 {\sqrt[4]{x}}^{2} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.1.3

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

$f^{- 1} (\sqrt[4]{x} + 4) = x^{\frac{4}{4}} + 4 {\sqrt[4]{x}}^{3} \cdot 4 + 6 {\sqrt[4]{x}}^{2} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.1.4

Cancel the common factor of $4$ .

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Step 5.2.3.2.1.4.1

Cancel the common factor.

Step 5.2.3.2.1.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 4 {\sqrt[4]{x}}^{3} \cdot 4 + 6 {\sqrt[4]{x}}^{2} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.1.5

Simplify.

Step 5.2.3.2.2

Rewrite ${\sqrt[4]{x}}^{3}$ as $\sqrt[4]{x^{3}}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 4 \sqrt[4]{x^{3}} \cdot 4 + 6 {\sqrt[4]{x}}^{2} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.3

Multiply $4$ by $4$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 6 {\sqrt[4]{x}}^{2} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.4

Rewrite ${\sqrt[4]{x}}^{2}$ as $\sqrt{x}$ .

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Step 5.2.3.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 6 {(x^{\frac{1}{4}})}^{2} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.4.2

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

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 6 x^{\frac{1}{4} \cdot 2} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.4.3

Combine $\frac{1}{4}$ and $2$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 6 x^{\frac{2}{4}} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.4.4

Cancel the common factor of $2$ and $4$ .

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Step 5.2.3.2.4.4.1

Factor $2$ out of $2$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 6 x^{\frac{2 (1)}{4}} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.4.4.2

Cancel the common factors.

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Step 5.2.3.2.4.4.2.1

Factor $2$ out of $4$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 6 x^{\frac{2 \cdot 1}{2 \cdot 2}} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.4.4.2.2

Cancel the common factor.

Step 5.2.3.2.4.4.2.3

Rewrite the expression.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 6 x^{\frac{1}{2}} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.4.5

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 6 \sqrt{x} \cdot 4^{2} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.5

Raise $4$ to the power of $2$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 6 \sqrt{x} \cdot 16 + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.6

Multiply $16$ by $6$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 4 \sqrt[4]{x} \cdot 4^{3} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.7

Multiply $4$ by $4^{3}$ by adding the exponents.

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Step 5.2.3.2.7.1

Move $4^{3}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 4^{3} \cdot (4 \sqrt[4]{x}) + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.7.2

Multiply $4^{3}$ by $4$ .

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Step 5.2.3.2.7.2.1

Raise $4$ to the power of $1$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 4^{3} \cdot (4 \sqrt[4]{x}) + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 4^{3 + 1} \sqrt[4]{x} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.7.3

Add $3$ and $1$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 4^{4} \sqrt[4]{x} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.8

Raise $4$ to the power of $4$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 4^{4} - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.2.9

Raise $4$ to the power of $4$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 {(\sqrt[4]{x} + 4)}^{3} + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.3

Use the Binomial Theorem.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 ({\sqrt[4]{x}}^{3} + 3 {\sqrt[4]{x}}^{2} \cdot 4 + 3 \sqrt[4]{x} \cdot 4^{2} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4

Simplify each term.

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Step 5.2.3.4.1

Rewrite ${\sqrt[4]{x}}^{3}$ as $\sqrt[4]{x^{3}}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 3 {\sqrt[4]{x}}^{2} \cdot 4 + 3 \sqrt[4]{x} \cdot 4^{2} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.2

Rewrite ${\sqrt[4]{x}}^{2}$ as $\sqrt{x}$ .

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Step 5.2.3.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 3 {(x^{\frac{1}{4}})}^{2} \cdot 4 + 3 \sqrt[4]{x} \cdot 4^{2} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.2.2

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

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 3 x^{\frac{1}{4} \cdot 2} \cdot 4 + 3 \sqrt[4]{x} \cdot 4^{2} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.2.3

Combine $\frac{1}{4}$ and $2$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 3 x^{\frac{2}{4}} \cdot 4 + 3 \sqrt[4]{x} \cdot 4^{2} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.2.4

Cancel the common factor of $2$ and $4$ .

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Step 5.2.3.4.2.4.1

Factor $2$ out of $2$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 3 x^{\frac{2 (1)}{4}} \cdot 4 + 3 \sqrt[4]{x} \cdot 4^{2} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.2.4.2

Cancel the common factors.

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Step 5.2.3.4.2.4.2.1

Factor $2$ out of $4$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 3 x^{\frac{2 \cdot 1}{2 \cdot 2}} \cdot 4 + 3 \sqrt[4]{x} \cdot 4^{2} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.2.4.2.2

Cancel the common factor.

Step 5.2.3.4.2.4.2.3

Rewrite the expression.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 3 x^{\frac{1}{2}} \cdot 4 + 3 \sqrt[4]{x} \cdot 4^{2} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.2.5

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 3 \sqrt{x} \cdot 4 + 3 \sqrt[4]{x} \cdot 4^{2} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.3

Multiply $4$ by $3$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 12 \sqrt{x} + 3 \sqrt[4]{x} \cdot 4^{2} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.4

Raise $4$ to the power of $2$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 12 \sqrt{x} + 3 \sqrt[4]{x} \cdot 16 + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.5

Multiply $16$ by $3$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 12 \sqrt{x} + 48 \sqrt[4]{x} + 4^{3}) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.4.6

Raise $4$ to the power of $3$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 (\sqrt[4]{x^{3}} + 12 \sqrt{x} + 48 \sqrt[4]{x} + 64) + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.5

Apply the distributive property.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 16 (12 \sqrt{x}) - 16 (48 \sqrt[4]{x}) - 16 \cdot 64 + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.6

Simplify.

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Step 5.2.3.6.1

Multiply $12$ by $- 16$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 16 (48 \sqrt[4]{x}) - 16 \cdot 64 + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.6.2

Multiply $48$ by $- 16$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 16 \cdot 64 + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.6.3

Multiply $- 16$ by $64$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 {(\sqrt[4]{x} + 4)}^{2} - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.7

Rewrite ${(\sqrt[4]{x} + 4)}^{2}$ as $(\sqrt[4]{x} + 4) (\sqrt[4]{x} + 4)$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 ((\sqrt[4]{x} + 4) (\sqrt[4]{x} + 4)) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.8

Expand $(\sqrt[4]{x} + 4) (\sqrt[4]{x} + 4)$ using the FOIL Method.

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Step 5.2.3.8.1

Apply the distributive property.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (\sqrt[4]{x} (\sqrt[4]{x} + 4) + 4 (\sqrt[4]{x} + 4)) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.8.2

Apply the distributive property.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (\sqrt[4]{x} \sqrt[4]{x} + \sqrt[4]{x} \cdot 4 + 4 (\sqrt[4]{x} + 4)) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.8.3

Apply the distributive property.

Step 5.2.3.9

Simplify and combine like terms.

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Step 5.2.3.9.1

Simplify each term.

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Step 5.2.3.9.1.1

Multiply $\sqrt[4]{x} \sqrt[4]{x}$ .

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Step 5.2.3.9.1.1.1

Raise $\sqrt[4]{x}$ to the power of $1$ .

Step 5.2.3.9.1.1.2

Raise $\sqrt[4]{x}$ to the power of $1$ .

Step 5.2.3.9.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 ({\sqrt[4]{x}}^{1 + 1} + \sqrt[4]{x} \cdot 4 + 4 \sqrt[4]{x} + 4 \cdot 4) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 ({\sqrt[4]{x}}^{2} + \sqrt[4]{x} \cdot 4 + 4 \sqrt[4]{x} + 4 \cdot 4) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.1.2

Rewrite ${\sqrt[4]{x}}^{2}$ as $\sqrt{x}$ .

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Step 5.2.3.9.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 ({(x^{\frac{1}{4}})}^{2} + \sqrt[4]{x} \cdot 4 + 4 \sqrt[4]{x} + 4 \cdot 4) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.1.2.2

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

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (x^{\frac{1}{4} \cdot 2} + \sqrt[4]{x} \cdot 4 + 4 \sqrt[4]{x} + 4 \cdot 4) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.1.2.3

Combine $\frac{1}{4}$ and $2$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (x^{\frac{2}{4}} + \sqrt[4]{x} \cdot 4 + 4 \sqrt[4]{x} + 4 \cdot 4) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.1.2.4

Cancel the common factor of $2$ and $4$ .

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Step 5.2.3.9.1.2.4.1

Factor $2$ out of $2$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (x^{\frac{2 (1)}{4}} + \sqrt[4]{x} \cdot 4 + 4 \sqrt[4]{x} + 4 \cdot 4) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.1.2.4.2

Cancel the common factors.

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Step 5.2.3.9.1.2.4.2.1

Factor $2$ out of $4$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (x^{\frac{2 \cdot 1}{2 \cdot 2}} + \sqrt[4]{x} \cdot 4 + 4 \sqrt[4]{x} + 4 \cdot 4) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.1.2.4.2.2

Cancel the common factor.

Step 5.2.3.9.1.2.4.2.3

Rewrite the expression.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (x^{\frac{1}{2}} + \sqrt[4]{x} \cdot 4 + 4 \sqrt[4]{x} + 4 \cdot 4) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.1.2.5

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (\sqrt{x} + \sqrt[4]{x} \cdot 4 + 4 \sqrt[4]{x} + 4 \cdot 4) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.1.3

Move $4$ to the left of $\sqrt[4]{x}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (\sqrt{x} + 4 \cdot \sqrt[4]{x} + 4 \sqrt[4]{x} + 4 \cdot 4) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.1.4

Multiply $4$ by $4$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (\sqrt{x} + 4 \sqrt[4]{x} + 4 \sqrt[4]{x} + 16) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.9.2

Add $4 \sqrt[4]{x}$ and $4 \sqrt[4]{x}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 (\sqrt{x} + 8 \sqrt[4]{x} + 16) - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.10

Apply the distributive property.

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 \sqrt{x} + 96 (8 \sqrt[4]{x}) + 96 \cdot 16 - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.11

Simplify.

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Step 5.2.3.11.1

Multiply $8$ by $96$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 \sqrt{x} + 768 \sqrt[4]{x} + 96 \cdot 16 - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.11.2

Multiply $96$ by $16$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 \sqrt{x} + 768 \sqrt[4]{x} + 1536 - 256 (\sqrt[4]{x} + 4) + 256$

Step 5.2.3.12

Apply the distributive property.

Step 5.2.3.13

Multiply $- 256$ by $4$ .

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x + 16 \sqrt[4]{x^{3}} + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 16 \sqrt[4]{x^{3}} - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 \sqrt{x} + 768 \sqrt[4]{x} + 1536 - 256 \sqrt[4]{x} - 1024 + 256$ .

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Step 5.2.4.1.1

Subtract $16 \sqrt[4]{x^{3}}$ from $16 \sqrt[4]{x^{3}}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 0 + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 \sqrt{x} + 768 \sqrt[4]{x} + 1536 - 256 \sqrt[4]{x} - 1024 + 256$

Step 5.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 192 \sqrt{x} - 768 \sqrt[4]{x} - 1024 + 96 \sqrt{x} + 768 \sqrt[4]{x} + 1536 - 256 \sqrt[4]{x} - 1024 + 256$

Step 5.2.4.1.3

Add $- 768 \sqrt[4]{x}$ and $768 \sqrt[4]{x}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 192 \sqrt{x} + 0 - 1024 + 96 \sqrt{x} + 1536 - 256 \sqrt[4]{x} - 1024 + 256$

Step 5.2.4.1.4

Add $x + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 192 \sqrt{x}$ and $0$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 96 \sqrt{x} + 256 \sqrt[4]{x} + 256 - 192 \sqrt{x} - 1024 + 96 \sqrt{x} + 1536 - 256 \sqrt[4]{x} - 1024 + 256$

Step 5.2.4.1.5

Subtract $256 \sqrt[4]{x}$ from $256 \sqrt[4]{x}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 96 \sqrt{x} + 0 + 256 - 192 \sqrt{x} - 1024 + 96 \sqrt{x} + 1536 - 1024 + 256$

Step 5.2.4.1.6

Add $x + 96 \sqrt{x}$ and $0$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 96 \sqrt{x} + 256 - 192 \sqrt{x} - 1024 + 96 \sqrt{x} + 1536 - 1024 + 256$

Step 5.2.4.2

Subtract $192 \sqrt{x}$ from $96 \sqrt{x}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 256 - 96 \sqrt{x} - 1024 + 96 \sqrt{x} + 1536 - 1024 + 256$

Step 5.2.4.3

Combine the opposite terms in $x + 256 - 96 \sqrt{x} - 1024 + 96 \sqrt{x} + 1536 - 1024 + 256$ .

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Step 5.2.4.3.1

Add $- 96 \sqrt{x}$ and $96 \sqrt{x}$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 256 + 0 - 1024 + 1536 - 1024 + 256$

Step 5.2.4.3.2

Add $x + 256$ and $0$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 256 - 1024 + 1536 - 1024 + 256$

Step 5.2.4.4

Simplify by adding and subtracting.

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Step 5.2.4.4.1

Subtract $1024$ from $256$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x - 768 + 1536 - 1024 + 256$

Step 5.2.4.4.2

Add $- 768$ and $1536$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 768 - 1024 + 256$

Step 5.2.4.4.3

Subtract $1024$ from $768$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x - 256 + 256$

Step 5.2.4.5

Combine the opposite terms in $x - 256 + 256$ .

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Step 5.2.4.5.1

Add $- 256$ and $256$ .

$f^{- 1} (\sqrt[4]{x} + 4) = x + 0$

Step 5.2.4.5.2

Add $x$ and $0$ .

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

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 $f (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256) = \sqrt[4]{x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256} + 4$

Step 5.3.3

Remove parentheses.

$f (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256) = \sqrt[4]{x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256} + 4$

Step 5.3.4

Simplify each term.

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Step 5.3.4.1

Make each term match the terms from the binomial theorem formula.

$f (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256) = \sqrt[4]{x^{4} - 4 \cdot (4 x^{3}) + 6 \cdot (4^{2} x^{2}) - 4 \cdot (4^{3} x) + 4^{4}} + 4$

Step 5.3.4.2

Factor $x^{4} - 4 \cdot 4 x^{3} + 6 \cdot 4^{2} x^{2} - 4 \cdot 4^{3} x + 4^{4}$ using the binomial theorem.

$f (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256) = \sqrt[4]{{(4 - x)}^{4}} + 4$

Step 5.3.4.3

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

$f (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256) = 4 - x + 4$

Step 5.3.5

Add $4$ and $4$ .

$f (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256) = - x + 8$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256$ is the inverse of $f (x) = \sqrt[4]{x} + 4$ .

$f^{- 1} (x) = x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256$