Precalculus Examples

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

Step 1

Write $f (x) = \sqrt[4]{x} - 3$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[4]{y} - 3 = x$ .

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

Step 3.2

Add $3$ to both sides of the equation.

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

Step 3.3

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

${\sqrt[4]{y}}^{4} = {(x + 3)}^{4}$

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 $y^{\frac{1}{4}}$ .

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

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.

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

Step 3.4.2.1.1.2.2

Rewrite the expression.

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

Step 3.4.2.1.2

Simplify.

$y = {(x + 3)}^{4}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x + 3)}^{4}$ .

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Step 3.4.3.1.1

Use the Binomial Theorem.

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

Step 3.4.3.1.2

Simplify each term.

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Step 3.4.3.1.2.1

Multiply $3$ by $4$ .

$y = x^{4} + 12 x^{3} + 6 x^{2} \cdot 3^{2} + 4 x \cdot 3^{3} + 3^{4}$

Step 3.4.3.1.2.2

Raise $3$ to the power of $2$ .

$y = x^{4} + 12 x^{3} + 6 x^{2} \cdot 9 + 4 x \cdot 3^{3} + 3^{4}$

Step 3.4.3.1.2.3

Multiply $9$ by $6$ .

$y = x^{4} + 12 x^{3} + 54 x^{2} + 4 x \cdot 3^{3} + 3^{4}$

Step 3.4.3.1.2.4

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

$y = x^{4} + 12 x^{3} + 54 x^{2} + 4 x \cdot 27 + 3^{4}$

Step 3.4.3.1.2.5

Multiply $27$ by $4$ .

$y = x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 3^{4}$

Step 3.4.3.1.2.6

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

$y = x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81$

Step 4

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

$f^{- 1} (x) = x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81$

Step 5

Verify if $f^{- 1} (x) = x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81$ is the inverse of $f (x) = \sqrt[4]{x} - 3$ .

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

$f^{- 1} (\sqrt[4]{x} - 3) = {(\sqrt[4]{x} - 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Use the Binomial Theorem.

$f^{- 1} (\sqrt[4]{x} - 3) = {\sqrt[4]{x}}^{4} + 4 {\sqrt[4]{x}}^{3} \cdot - 3 + 6 {\sqrt[4]{x}}^{2} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

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 $x^{\frac{1}{4}}$ .

$f^{- 1} (\sqrt[4]{x} - 3) = {(x^{\frac{1}{4}})}^{4} + 4 {\sqrt[4]{x}}^{3} \cdot - 3 + 6 {\sqrt[4]{x}}^{2} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.1.2

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

$f^{- 1} (\sqrt[4]{x} - 3) = x^{\frac{1}{4} \cdot 4} + 4 {\sqrt[4]{x}}^{3} \cdot - 3 + 6 {\sqrt[4]{x}}^{2} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.1.3

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

$f^{- 1} (\sqrt[4]{x} - 3) = x^{\frac{4}{4}} + 4 {\sqrt[4]{x}}^{3} \cdot - 3 + 6 {\sqrt[4]{x}}^{2} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

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.

$f^{- 1} (\sqrt[4]{x} - 3) = x + 4 {\sqrt[4]{x}}^{3} \cdot - 3 + 6 {\sqrt[4]{x}}^{2} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.1.5

Simplify.

Step 5.2.3.2.2

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

$f^{- 1} (\sqrt[4]{x} - 3) = x + 4 \sqrt[4]{x^{3}} \cdot - 3 + 6 {\sqrt[4]{x}}^{2} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.3

Multiply $- 3$ by $4$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 6 {\sqrt[4]{x}}^{2} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

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 $x^{\frac{1}{4}}$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 6 {(x^{\frac{1}{4}})}^{2} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.4.2

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 6 x^{\frac{1}{4} \cdot 2} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.4.3

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 6 x^{\frac{2}{4}} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

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 $2$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 6 x^{\frac{2 (1)}{4}} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

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 $4$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 6 x^{\frac{2 \cdot 1}{2 \cdot 2}} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

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.

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 6 x^{\frac{1}{2}} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.4.5

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 6 \sqrt{x} {(- 3)}^{2} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.5

Raise $- 3$ to the power of $2$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 6 \sqrt{x} \cdot 9 + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.6

Multiply $9$ by $6$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} + 4 \sqrt[4]{x} {(- 3)}^{3} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.7

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} + 4 \sqrt[4]{x} \cdot - 27 + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.8

Multiply $- 27$ by $4$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + {(- 3)}^{4} + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.2.9

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 {(\sqrt[4]{x} - 3)}^{3} + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.3

Use the Binomial Theorem.

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 ({\sqrt[4]{x}}^{3} + 3 {\sqrt[4]{x}}^{2} \cdot - 3 + 3 \sqrt[4]{x} {(- 3)}^{2} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.4

Simplify each term.

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Step 5.2.3.4.1

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} + 3 {\sqrt[4]{x}}^{2} \cdot - 3 + 3 \sqrt[4]{x} {(- 3)}^{2} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

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 $x^{\frac{1}{4}}$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} + 3 {(x^{\frac{1}{4}})}^{2} \cdot - 3 + 3 \sqrt[4]{x} {(- 3)}^{2} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.4.2.2

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} + 3 x^{\frac{1}{4} \cdot 2} \cdot - 3 + 3 \sqrt[4]{x} {(- 3)}^{2} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.4.2.3

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} + 3 x^{\frac{2}{4}} \cdot - 3 + 3 \sqrt[4]{x} {(- 3)}^{2} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

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 $2$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} + 3 x^{\frac{2 (1)}{4}} \cdot - 3 + 3 \sqrt[4]{x} {(- 3)}^{2} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

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 $4$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} + 3 x^{\frac{2 \cdot 1}{2 \cdot 2}} \cdot - 3 + 3 \sqrt[4]{x} {(- 3)}^{2} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

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.

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} + 3 x^{\frac{1}{2}} \cdot - 3 + 3 \sqrt[4]{x} {(- 3)}^{2} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.4.2.5

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} + 3 \sqrt{x} \cdot - 3 + 3 \sqrt[4]{x} {(- 3)}^{2} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.4.3

Multiply $- 3$ by $3$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} - 9 \sqrt{x} + 3 \sqrt[4]{x} {(- 3)}^{2} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.4.4

Raise $- 3$ to the power of $2$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} - 9 \sqrt{x} + 3 \sqrt[4]{x} \cdot 9 + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.4.5

Multiply $9$ by $3$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} - 9 \sqrt{x} + 27 \sqrt[4]{x} + {(- 3)}^{3}) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.4.6

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 (\sqrt[4]{x^{3}} - 9 \sqrt{x} + 27 \sqrt[4]{x} - 27) + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.5

Apply the distributive property.

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} + 12 (- 9 \sqrt{x}) + 12 (27 \sqrt[4]{x}) + 12 \cdot - 27 + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.6

Simplify.

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Step 5.2.3.6.1

Multiply $- 9$ by $12$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 12 (27 \sqrt[4]{x}) + 12 \cdot - 27 + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.6.2

Multiply $27$ by $12$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} + 12 \cdot - 27 + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.6.3

Multiply $12$ by $- 27$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 {(\sqrt[4]{x} - 3)}^{2} + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.7

Rewrite ${(\sqrt[4]{x} - 3)}^{2}$ as $(\sqrt[4]{x} - 3) (\sqrt[4]{x} - 3)$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 ((\sqrt[4]{x} - 3) (\sqrt[4]{x} - 3)) + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.8

Expand $(\sqrt[4]{x} - 3) (\sqrt[4]{x} - 3)$ using the FOIL Method.

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Step 5.2.3.8.1

Apply the distributive property.

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (\sqrt[4]{x} (\sqrt[4]{x} - 3) - 3 (\sqrt[4]{x} - 3)) + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.8.2

Apply the distributive property.

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (\sqrt[4]{x} \sqrt[4]{x} + \sqrt[4]{x} \cdot - 3 - 3 (\sqrt[4]{x} - 3)) + 108 (\sqrt[4]{x} - 3) + 81$

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.

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 ({\sqrt[4]{x}}^{1 + 1} + \sqrt[4]{x} \cdot - 3 - 3 \sqrt[4]{x} - 3 \cdot - 3) + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.9.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 ({\sqrt[4]{x}}^{2} + \sqrt[4]{x} \cdot - 3 - 3 \sqrt[4]{x} - 3 \cdot - 3) + 108 (\sqrt[4]{x} - 3) + 81$

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 $x^{\frac{1}{4}}$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 ({(x^{\frac{1}{4}})}^{2} + \sqrt[4]{x} \cdot - 3 - 3 \sqrt[4]{x} - 3 \cdot - 3) + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.9.1.2.2

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (x^{\frac{1}{4} \cdot 2} + \sqrt[4]{x} \cdot - 3 - 3 \sqrt[4]{x} - 3 \cdot - 3) + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.9.1.2.3

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (x^{\frac{2}{4}} + \sqrt[4]{x} \cdot - 3 - 3 \sqrt[4]{x} - 3 \cdot - 3) + 108 (\sqrt[4]{x} - 3) + 81$

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 $2$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (x^{\frac{2 (1)}{4}} + \sqrt[4]{x} \cdot - 3 - 3 \sqrt[4]{x} - 3 \cdot - 3) + 108 (\sqrt[4]{x} - 3) + 81$

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 $4$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (x^{\frac{2 \cdot 1}{2 \cdot 2}} + \sqrt[4]{x} \cdot - 3 - 3 \sqrt[4]{x} - 3 \cdot - 3) + 108 (\sqrt[4]{x} - 3) + 81$

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.

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (x^{\frac{1}{2}} + \sqrt[4]{x} \cdot - 3 - 3 \sqrt[4]{x} - 3 \cdot - 3) + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.9.1.2.5

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (\sqrt{x} + \sqrt[4]{x} \cdot - 3 - 3 \sqrt[4]{x} - 3 \cdot - 3) + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.9.1.3

Move $- 3$ to the left of $\sqrt[4]{x}$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (\sqrt{x} - 3 \cdot \sqrt[4]{x} - 3 \sqrt[4]{x} - 3 \cdot - 3) + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.9.1.4

Multiply $- 3$ by $- 3$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (\sqrt{x} - 3 \sqrt[4]{x} - 3 \sqrt[4]{x} + 9) + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.9.2

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

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 (\sqrt{x} - 6 \sqrt[4]{x} + 9) + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.10

Apply the distributive property.

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 \sqrt{x} + 54 (- 6 \sqrt[4]{x}) + 54 \cdot 9 + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.11

Simplify.

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Step 5.2.3.11.1

Multiply $- 6$ by $54$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 \sqrt{x} - 324 \sqrt[4]{x} + 54 \cdot 9 + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.11.2

Multiply $54$ by $9$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 \sqrt{x} - 324 \sqrt[4]{x} + 486 + 108 (\sqrt[4]{x} - 3) + 81$

Step 5.2.3.12

Apply the distributive property.

Step 5.2.3.13

Multiply $108$ by $- 3$ .

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x - 12 \sqrt[4]{x^{3}} + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 + 12 \sqrt[4]{x^{3}} - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 \sqrt{x} - 324 \sqrt[4]{x} + 486 + 108 \sqrt[4]{x} - 324 + 81$ .

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Step 5.2.4.1.1

Add $- 12 \sqrt[4]{x^{3}}$ and $12 \sqrt[4]{x^{3}}$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 0 + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 \sqrt{x} - 324 \sqrt[4]{x} + 486 + 108 \sqrt[4]{x} - 324 + 81$

Step 5.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 - 108 \sqrt{x} + 324 \sqrt[4]{x} - 324 + 54 \sqrt{x} - 324 \sqrt[4]{x} + 486 + 108 \sqrt[4]{x} - 324 + 81$

Step 5.2.4.1.3

Subtract $324 \sqrt[4]{x}$ from $324 \sqrt[4]{x}$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 - 108 \sqrt{x} + 0 - 324 + 54 \sqrt{x} + 486 + 108 \sqrt[4]{x} - 324 + 81$

Step 5.2.4.1.4

Add $x + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 - 108 \sqrt{x}$ and $0$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 54 \sqrt{x} - 108 \sqrt[4]{x} + 81 - 108 \sqrt{x} - 324 + 54 \sqrt{x} + 486 + 108 \sqrt[4]{x} - 324 + 81$

Step 5.2.4.1.5

Add $- 108 \sqrt[4]{x}$ and $108 \sqrt[4]{x}$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 54 \sqrt{x} + 0 + 81 - 108 \sqrt{x} - 324 + 54 \sqrt{x} + 486 - 324 + 81$

Step 5.2.4.1.6

Add $x + 54 \sqrt{x}$ and $0$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 54 \sqrt{x} + 81 - 108 \sqrt{x} - 324 + 54 \sqrt{x} + 486 - 324 + 81$

Step 5.2.4.2

Subtract $108 \sqrt{x}$ from $54 \sqrt{x}$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 81 - 54 \sqrt{x} - 324 + 54 \sqrt{x} + 486 - 324 + 81$

Step 5.2.4.3

Combine the opposite terms in $x + 81 - 54 \sqrt{x} - 324 + 54 \sqrt{x} + 486 - 324 + 81$ .

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Step 5.2.4.3.1

Add $- 54 \sqrt{x}$ and $54 \sqrt{x}$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 81 + 0 - 324 + 486 - 324 + 81$

Step 5.2.4.3.2

Add $x + 81$ and $0$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 81 - 324 + 486 - 324 + 81$

Step 5.2.4.4

Simplify by adding and subtracting.

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Step 5.2.4.4.1

Subtract $324$ from $81$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 243 + 486 - 324 + 81$

Step 5.2.4.4.2

Add $- 243$ and $486$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 243 - 324 + 81$

Step 5.2.4.4.3

Subtract $324$ from $243$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x - 81 + 81$

Step 5.2.4.5

Combine the opposite terms in $x - 81 + 81$ .

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Step 5.2.4.5.1

Add $- 81$ and $81$ .

$f^{- 1} (\sqrt[4]{x} - 3) = x + 0$

Step 5.2.4.5.2

Add $x$ and $0$ .

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

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 $f (x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81) = \sqrt[4]{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} - 3$

Step 5.3.3

Remove parentheses.

$f (x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81) = \sqrt[4]{x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81} - 3$

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.

$f (x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81) = \sqrt[4]{x^{4} + 4 \cdot (3 x^{3}) + 6 \cdot (3^{2} x^{2}) + 4 \cdot (3^{3} x) + 3^{4}} - 3$

Step 5.3.4.2

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

$f (x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81) = \sqrt[4]{{(x + 3)}^{4}} - 3$

Step 5.3.4.3

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

$f (x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81) = x + 3 - 3$

Step 5.3.5

Combine the opposite terms in $x + 3 - 3$ .

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Step 5.3.5.1

Subtract $3$ from $3$ .

$f (x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81) = x + 0$

Step 5.3.5.2

Add $x$ and $0$ .

$f (x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81$ is the inverse of $f (x) = \sqrt[4]{x} - 3$ .

$f^{- 1} (x) = x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81$