Precalculus Examples

$f (x) = \frac{\sqrt[3]{x} - 7}{5}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \frac{\sqrt[3]{x} - 7}{5}$ as an equation.

$y = \frac{\sqrt[3]{x} - 7}{5}$

Step 2

Interchange the variables.

$x = \frac{\sqrt[3]{y} - 7}{5}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{\sqrt[3]{y} - 7}{5} = x$ .

$\frac{\sqrt[3]{y} - 7}{5} = x$

Step 3.2

Multiply both sides of the equation by $5$ .

$5 \frac{\sqrt[3]{y} - 7}{5} = 5 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $5$ .

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Step 3.3.1.1

Cancel the common factor.

$5 \frac{\sqrt[3]{y} - 7}{5} = 5 x$

Step 3.3.1.2

Rewrite the expression.

$\sqrt[3]{y} - 7 = 5 x$

Step 3.4

Add $7$ to both sides of the equation.

$\sqrt[3]{y} = 5 x + 7$

Step 3.5

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{y}}^{3} = {(5 x + 7)}^{3}$

Step 3.6

Simplify each side of the equation.

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Step 3.6.1

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

${(y^{\frac{1}{3}})}^{3} = {(5 x + 7)}^{3}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Simplify ${(y^{\frac{1}{3}})}^{3}$ .

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Step 3.6.2.1.1

Multiply the exponents in ${(y^{\frac{1}{3}})}^{3}$ .

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Step 3.6.2.1.1.1

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

$y^{\frac{1}{3} \cdot 3} = {(5 x + 7)}^{3}$

Step 3.6.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.6.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(5 x + 7)}^{3}$

Step 3.6.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(5 x + 7)}^{3}$

Step 3.6.2.1.2

Simplify.

$y = {(5 x + 7)}^{3}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Simplify ${(5 x + 7)}^{3}$ .

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Step 3.6.3.1.1

Use the Binomial Theorem.

$y = {(5 x)}^{3} + 3 {(5 x)}^{2} \cdot 7 + 3 (5 x) \cdot 7^{2} + 7^{3}$

Step 3.6.3.1.2

Simplify each term.

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Step 3.6.3.1.2.1

Apply the product rule to $5 x$ .

$y = 5^{3} x^{3} + 3 {(5 x)}^{2} \cdot 7 + 3 (5 x) \cdot 7^{2} + 7^{3}$

Step 3.6.3.1.2.2

Raise $5$ to the power of $3$ .

$y = 125 x^{3} + 3 {(5 x)}^{2} \cdot 7 + 3 (5 x) \cdot 7^{2} + 7^{3}$

Step 3.6.3.1.2.3

Apply the product rule to $5 x$ .

$y = 125 x^{3} + 3 (5^{2} x^{2}) \cdot 7 + 3 (5 x) \cdot 7^{2} + 7^{3}$

Step 3.6.3.1.2.4

Raise $5$ to the power of $2$ .

$y = 125 x^{3} + 3 (25 x^{2}) \cdot 7 + 3 (5 x) \cdot 7^{2} + 7^{3}$

Step 3.6.3.1.2.5

Multiply $25$ by $3$ .

$y = 125 x^{3} + 75 x^{2} \cdot 7 + 3 (5 x) \cdot 7^{2} + 7^{3}$

Step 3.6.3.1.2.6

Multiply $7$ by $75$ .

$y = 125 x^{3} + 525 x^{2} + 3 (5 x) \cdot 7^{2} + 7^{3}$

Step 3.6.3.1.2.7

Multiply $5$ by $3$ .

$y = 125 x^{3} + 525 x^{2} + 15 x \cdot 7^{2} + 7^{3}$

Step 3.6.3.1.2.8

Raise $7$ to the power of $2$ .

$y = 125 x^{3} + 525 x^{2} + 15 x \cdot 49 + 7^{3}$

Step 3.6.3.1.2.9

Multiply $49$ by $15$ .

$y = 125 x^{3} + 525 x^{2} + 735 x + 7^{3}$

Step 3.6.3.1.2.10

Raise $7$ to the power of $3$ .

$y = 125 x^{3} + 525 x^{2} + 735 x + 343$

Step 4

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

$f^{- 1} (x) = 125 x^{3} + 525 x^{2} + 735 x + 343$

Step 5

Verify if $f^{- 1} (x) = 125 x^{3} + 525 x^{2} + 735 x + 343$ is the inverse of $f (x) = \frac{\sqrt[3]{x} - 7}{5}$ .

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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 $f^{- 1} (\frac{\sqrt[3]{x} - 7}{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = 125 {(\frac{\sqrt[3]{x} - 7}{5})}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Apply the product rule to $\frac{\sqrt[3]{x} - 7}{5}$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = 125 (\frac{{(\sqrt[3]{x} - 7)}^{3}}{5^{3}}) + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.2

Raise $5$ to the power of $3$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = 125 (\frac{{(\sqrt[3]{x} - 7)}^{3}}{125}) + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.3

Cancel the common factor of $125$ .

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Step 5.2.3.3.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = 125 (\frac{{(\sqrt[3]{x} - 7)}^{3}}{125}) + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.3.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = {(\sqrt[3]{x} - 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.4

Use the Binomial Theorem.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = {\sqrt[3]{x}}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.5

Simplify each term.

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Step 5.2.3.5.1

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

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Step 5.2.3.5.1.1

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

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = {(x^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.5.1.2

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

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.5.1.3

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

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.5.1.4

Cancel the common factor of $3$ .

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Step 5.2.3.5.1.4.1

Cancel the common factor.

Step 5.2.3.5.1.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.5.1.5

Simplify.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x + 3 {\sqrt[3]{x}}^{2} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.5.2

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x + 3 \sqrt[3]{x^{2}} \cdot - 7 + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.5.3

Multiply $- 7$ by $3$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} {(- 7)}^{2} + {(- 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.5.4

Raise $- 7$ to the power of $2$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 49 + {(- 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.5.5

Multiply $49$ by $3$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} + {(- 7)}^{3} + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.5.6

Raise $- 7$ to the power of $3$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 525 {(\frac{\sqrt[3]{x} - 7}{5})}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.6

Apply the product rule to $\frac{\sqrt[3]{x} - 7}{5}$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 525 (\frac{{(\sqrt[3]{x} - 7)}^{2}}{5^{2}}) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.7

Raise $5$ to the power of $2$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 525 (\frac{{(\sqrt[3]{x} - 7)}^{2}}{25}) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.8

Cancel the common factor of $25$ .

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Step 5.2.3.8.1

Factor $25$ out of $525$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 25 (21) (\frac{{(\sqrt[3]{x} - 7)}^{2}}{25}) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.8.2

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 25 \cdot (21 (\frac{{(\sqrt[3]{x} - 7)}^{2}}{25})) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.8.3

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 {(\sqrt[3]{x} - 7)}^{2} + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.9

Rewrite ${(\sqrt[3]{x} - 7)}^{2}$ as $(\sqrt[3]{x} - 7) (\sqrt[3]{x} - 7)$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 ((\sqrt[3]{x} - 7) (\sqrt[3]{x} - 7)) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.10

Expand $(\sqrt[3]{x} - 7) (\sqrt[3]{x} - 7)$ using the FOIL Method.

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Step 5.2.3.10.1

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x} (\sqrt[3]{x} - 7) - 7 (\sqrt[3]{x} - 7)) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.10.2

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 7 - 7 (\sqrt[3]{x} - 7)) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.10.3

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot - 7 - 7 \sqrt[3]{x} - 7 \cdot - 7) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.11

Simplify and combine like terms.

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Step 5.2.3.11.1

Simplify each term.

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Step 5.2.3.11.1.1

Multiply $\sqrt[3]{x} \sqrt[3]{x}$ .

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Step 5.2.3.11.1.1.1

Raise $\sqrt[3]{x}$ to the power of $1$ .

Step 5.2.3.11.1.1.2

Raise $\sqrt[3]{x}$ to the power of $1$ .

Step 5.2.3.11.1.1.3

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

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 ({\sqrt[3]{x}}^{1 + 1} + \sqrt[3]{x} \cdot - 7 - 7 \sqrt[3]{x} - 7 \cdot - 7) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.11.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 ({\sqrt[3]{x}}^{2} + \sqrt[3]{x} \cdot - 7 - 7 \sqrt[3]{x} - 7 \cdot - 7) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.11.1.2

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot - 7 - 7 \sqrt[3]{x} - 7 \cdot - 7) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.11.1.3

Move $- 7$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x^{2}} - 7 \cdot \sqrt[3]{x} - 7 \sqrt[3]{x} - 7 \cdot - 7) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.11.1.4

Multiply $- 7$ by $- 7$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x^{2}} - 7 \sqrt[3]{x} - 7 \sqrt[3]{x} + 49) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.11.2

Subtract $7 \sqrt[3]{x}$ from $- 7 \sqrt[3]{x}$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 (\sqrt[3]{x^{2}} - 14 \sqrt[3]{x} + 49) + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.12

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} + 21 (- 14 \sqrt[3]{x}) + 21 \cdot 49 + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.13

Simplify.

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Step 5.2.3.13.1

Multiply $- 14$ by $21$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 21 \cdot 49 + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.13.2

Multiply $21$ by $49$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 735 (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.14

Cancel the common factor of $5$ .

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Step 5.2.3.14.1

Factor $5$ out of $735$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 5 (147) (\frac{\sqrt[3]{x} - 7}{5}) + 343$

Step 5.2.3.14.2

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 5 \cdot (147 (\frac{\sqrt[3]{x} - 7}{5})) + 343$

Step 5.2.3.14.3

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 147 (\sqrt[3]{x} - 7) + 343$

Step 5.2.3.15

Apply the distributive property.

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 147 \sqrt[3]{x} + 147 \cdot - 7 + 343$

Step 5.2.3.16

Multiply $147$ by $- 7$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 147 \sqrt[3]{x} - 1029 + 343$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x - 21 \sqrt[3]{x^{2}} + 147 \sqrt[3]{x} - 343 + 21 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 1029 + 147 \sqrt[3]{x} - 1029 + 343$ .

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Step 5.2.4.1.1

Add $- 21 \sqrt[3]{x^{2}}$ and $21 \sqrt[3]{x^{2}}$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x + 0 + 147 \sqrt[3]{x} - 343 - 294 \sqrt[3]{x} + 1029 + 147 \sqrt[3]{x} - 1029 + 343$

Step 5.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x + 147 \sqrt[3]{x} - 343 - 294 \sqrt[3]{x} + 1029 + 147 \sqrt[3]{x} - 1029 + 343$

Step 5.2.4.1.3

Subtract $1029$ from $1029$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x + 147 \sqrt[3]{x} - 343 - 294 \sqrt[3]{x} + 0 + 147 \sqrt[3]{x} + 343$

Step 5.2.4.1.4

Add $x + 147 \sqrt[3]{x} - 343 - 294 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x + 147 \sqrt[3]{x} - 343 - 294 \sqrt[3]{x} + 147 \sqrt[3]{x} + 343$

Step 5.2.4.1.5

Add $- 343$ and $343$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x + 147 \sqrt[3]{x} + 0 - 294 \sqrt[3]{x} + 147 \sqrt[3]{x}$

Step 5.2.4.1.6

Add $x + 147 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x + 147 \sqrt[3]{x} - 294 \sqrt[3]{x} + 147 \sqrt[3]{x}$

Step 5.2.4.2

Subtract $294 \sqrt[3]{x}$ from $147 \sqrt[3]{x}$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x - 147 \sqrt[3]{x} + 147 \sqrt[3]{x}$

Step 5.2.4.3

Combine the opposite terms in $x - 147 \sqrt[3]{x} + 147 \sqrt[3]{x}$ .

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Step 5.2.4.3.1

Add $- 147 \sqrt[3]{x}$ and $147 \sqrt[3]{x}$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[3]{x} - 7}{5}) = x$

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 $f (125 x^{3} + 525 x^{2} + 735 x + 343)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (125 x^{3} + 525 x^{2} + 735 x + 343) = \frac{\sqrt[3]{125 x^{3} + 525 x^{2} + 735 x + 343} - 7}{5}$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 125 x^{3} + 525 x^{2} + 735 x + 343$ is the inverse of $f (x) = \frac{\sqrt[3]{x} - 7}{5}$ .

$f^{- 1} (x) = 125 x^{3} + 525 x^{2} + 735 x + 343$