Find the Inverse cube root of x+3-5

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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

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

Add $5$ to both sides of the equation.

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Use the Binomial Theorem.

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

Simplify each term.

Multiply $5$ by $3$ .

$y + 3 = x^{3} + 15 x^{2} + 3 x \cdot 5^{2} + 5^{3}$

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

$y + 3 = x^{3} + 15 x^{2} + 3 x \cdot 25 + 5^{3}$

Multiply $25$ by $3$ .

$y + 3 = x^{3} + 15 x^{2} + 75 x + 5^{3}$

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

$y + 3 = x^{3} + 15 x^{2} + 75 x + 125$

Move all terms not containing $y$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$y = x^{3} + 15 x^{2} + 75 x + 125 - 3$

Subtract $3$ from $125$ .

$y = x^{3} + 15 x^{2} + 75 x + 122$

Step 3

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

$f^{- 1} (x) = x^{3} + 15 x^{2} + 75 x + 122$

Step 4

Verify if $f^{- 1} (x) = x^{3} + 15 x^{2} + 75 x + 122$ is the inverse of $f (x) = \sqrt[3]{x + 3} - 5$ .

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

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

Set up the composite result function.

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

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

$f^{- 1} (\sqrt[3]{x + 3} - 5) = {(\sqrt[3]{x + 3} - 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

Simplify each term.

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{x + 3} - 5) = {\sqrt[3]{x + 3}}^{3} + 3 {\sqrt[3]{x + 3}}^{2} \cdot - 5 + 3 \sqrt[3]{x + 3} {(- 5)}^{2} + {(- 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

Simplify each term.

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

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

$f^{- 1} (\sqrt[3]{x + 3} - 5) = {({(x + 3)}^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x + 3}}^{2} \cdot - 5 + 3 \sqrt[3]{x + 3} {(- 5)}^{2} + {(- 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

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

$f^{- 1} (\sqrt[3]{x + 3} - 5) = {(x + 3)}^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x + 3}}^{2} \cdot - 5 + 3 \sqrt[3]{x + 3} {(- 5)}^{2} + {(- 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

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

$f^{- 1} (\sqrt[3]{x + 3} - 5) = {(x + 3)}^{\frac{3}{3}} + 3 {\sqrt[3]{x + 3}}^{2} \cdot - 5 + 3 \sqrt[3]{x + 3} {(- 5)}^{2} + {(- 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

Cancel the common factor of $3$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (\sqrt[3]{x + 3} - 5) = (x + 3) + 3 {\sqrt[3]{x + 3}}^{2} \cdot - 5 + 3 \sqrt[3]{x + 3} {(- 5)}^{2} + {(- 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

Simplify.

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x + 3 + 3 {\sqrt[3]{x + 3}}^{2} \cdot - 5 + 3 \sqrt[3]{x + 3} {(- 5)}^{2} + {(- 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

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

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x + 3 + 3 \sqrt[3]{{(x + 3)}^{2}} \cdot - 5 + 3 \sqrt[3]{x + 3} {(- 5)}^{2} + {(- 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

Multiply $- 5$ by $3$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x + 3 - 15 \sqrt[3]{{(x + 3)}^{2}} + 3 \sqrt[3]{x + 3} {(- 5)}^{2} + {(- 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

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

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x + 3 - 15 \sqrt[3]{{(x + 3)}^{2}} + 3 \sqrt[3]{x + 3} \cdot 25 + {(- 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

Multiply $25$ by $3$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x + 3 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + {(- 5)}^{3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

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

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x + 3 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} - 125 + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

Subtract $125$ from $3$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 {(\sqrt[3]{x + 3} - 5)}^{2} + 75 (\sqrt[3]{x + 3} - 5) + 122$

Rewrite ${(\sqrt[3]{x + 3} - 5)}^{2}$ as $(\sqrt[3]{x + 3} - 5) (\sqrt[3]{x + 3} - 5)$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 ((\sqrt[3]{x + 3} - 5) (\sqrt[3]{x + 3} - 5)) + 75 (\sqrt[3]{x + 3} - 5) + 122$

Expand $(\sqrt[3]{x + 3} - 5) (\sqrt[3]{x + 3} - 5)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 (\sqrt[3]{x + 3} (\sqrt[3]{x + 3} - 5) - 5 (\sqrt[3]{x + 3} - 5)) + 75 (\sqrt[3]{x + 3} - 5) + 122$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 (\sqrt[3]{x + 3} \sqrt[3]{x + 3} + \sqrt[3]{x + 3} \cdot - 5 - 5 (\sqrt[3]{x + 3} - 5)) + 75 (\sqrt[3]{x + 3} - 5) + 122$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 (\sqrt[3]{x + 3} \sqrt[3]{x + 3} + \sqrt[3]{x + 3} \cdot - 5 - 5 \sqrt[3]{x + 3} - 5 \cdot - 5) + 75 (\sqrt[3]{x + 3} - 5) + 122$

Simplify and combine like terms.

Simplify each term.

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

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

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

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 ({\sqrt[3]{x + 3}}^{1 + 1} + \sqrt[3]{x + 3} \cdot - 5 - 5 \sqrt[3]{x + 3} - 5 \cdot - 5) + 75 (\sqrt[3]{x + 3} - 5) + 122$

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 ({\sqrt[3]{x + 3}}^{2} + \sqrt[3]{x + 3} \cdot - 5 - 5 \sqrt[3]{x + 3} - 5 \cdot - 5) + 75 (\sqrt[3]{x + 3} - 5) + 122$

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

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 (\sqrt[3]{{(x + 3)}^{2}} + \sqrt[3]{x + 3} \cdot - 5 - 5 \sqrt[3]{x + 3} - 5 \cdot - 5) + 75 (\sqrt[3]{x + 3} - 5) + 122$

Move $- 5$ to the left of $\sqrt[3]{x + 3}$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 (\sqrt[3]{{(x + 3)}^{2}} - 5 \cdot \sqrt[3]{x + 3} - 5 \sqrt[3]{x + 3} - 5 \cdot - 5) + 75 (\sqrt[3]{x + 3} - 5) + 122$

Multiply $- 5$ by $- 5$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 (\sqrt[3]{{(x + 3)}^{2}} - 5 \sqrt[3]{x + 3} - 5 \sqrt[3]{x + 3} + 25) + 75 (\sqrt[3]{x + 3} - 5) + 122$

Subtract $5 \sqrt[3]{x + 3}$ from $- 5 \sqrt[3]{x + 3}$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 (\sqrt[3]{{(x + 3)}^{2}} - 10 \sqrt[3]{x + 3} + 25) + 75 (\sqrt[3]{x + 3} - 5) + 122$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 \sqrt[3]{{(x + 3)}^{2}} + 15 (- 10 \sqrt[3]{x + 3}) + 15 \cdot 25 + 75 (\sqrt[3]{x + 3} - 5) + 122$

Simplify.

Multiply $- 10$ by $15$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 \sqrt[3]{{(x + 3)}^{2}} - 150 \sqrt[3]{x + 3} + 15 \cdot 25 + 75 (\sqrt[3]{x + 3} - 5) + 122$

Multiply $15$ by $25$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 \sqrt[3]{{(x + 3)}^{2}} - 150 \sqrt[3]{x + 3} + 375 + 75 (\sqrt[3]{x + 3} - 5) + 122$

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 \sqrt[3]{{(x + 3)}^{2}} - 150 \sqrt[3]{x + 3} + 375 + 75 \sqrt[3]{x + 3} + 75 \cdot - 5 + 122$

Multiply $75$ by $- 5$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 \sqrt[3]{{(x + 3)}^{2}} - 150 \sqrt[3]{x + 3} + 375 + 75 \sqrt[3]{x + 3} - 375 + 122$

Simplify by adding terms.

Combine the opposite terms in $x - 122 - 15 \sqrt[3]{{(x + 3)}^{2}} + 75 \sqrt[3]{x + 3} + 15 \sqrt[3]{{(x + 3)}^{2}} - 150 \sqrt[3]{x + 3} + 375 + 75 \sqrt[3]{x + 3} - 375 + 122$ .

Add $- 15 \sqrt[3]{{(x + 3)}^{2}}$ and $15 \sqrt[3]{{(x + 3)}^{2}}$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 + 0 + 75 \sqrt[3]{x + 3} - 150 \sqrt[3]{x + 3} + 375 + 75 \sqrt[3]{x + 3} - 375 + 122$

Add $x - 122$ and $0$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 + 75 \sqrt[3]{x + 3} - 150 \sqrt[3]{x + 3} + 375 + 75 \sqrt[3]{x + 3} - 375 + 122$

Subtract $375$ from $375$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 + 75 \sqrt[3]{x + 3} - 150 \sqrt[3]{x + 3} + 0 + 75 \sqrt[3]{x + 3} + 122$

Add $x - 122 + 75 \sqrt[3]{x + 3} - 150 \sqrt[3]{x + 3}$ and $0$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 122 + 75 \sqrt[3]{x + 3} - 150 \sqrt[3]{x + 3} + 75 \sqrt[3]{x + 3} + 122$

Add $- 122$ and $122$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x + 0 + 75 \sqrt[3]{x + 3} - 150 \sqrt[3]{x + 3} + 75 \sqrt[3]{x + 3}$

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x + 75 \sqrt[3]{x + 3} - 150 \sqrt[3]{x + 3} + 75 \sqrt[3]{x + 3}$

Subtract $150 \sqrt[3]{x + 3}$ from $75 \sqrt[3]{x + 3}$ .

$f^{- 1} (\sqrt[3]{x + 3} - 5) = x - 75 \sqrt[3]{x + 3} + 75 \sqrt[3]{x + 3}$

Combine the opposite terms in $x - 75 \sqrt[3]{x + 3} + 75 \sqrt[3]{x + 3}$ .

Add $- 75 \sqrt[3]{x + 3}$ and $75 \sqrt[3]{x + 3}$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (x^{3} + 15 x^{2} + 75 x + 122)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x^{3} + 15 x^{2} + 75 x + 122) + 3} - 5$

Simplify each term.

Add $122$ and $3$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{x^{3} + 15 x^{2} + 75 x + 125} - 5$

Rewrite $x^{3} + 15 x^{2} + 75 x + 125$ in a factored form.

Regroup terms.

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{x^{3} + 125 + 15 x^{2} + 75 x} - 5$

Rewrite $125$ as $5^{3}$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{x^{3} + 5^{3} + 15 x^{2} + 75 x} - 5$

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 5$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} - x \cdot 5 + 5^{2}) + 15 x^{2} + 75 x} - 5$

Simplify.

Multiply $5$ by $- 1$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} - 5 x + 5^{2}) + 15 x^{2} + 75 x} - 5$

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

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} - 5 x + 25) + 15 x^{2} + 75 x} - 5$

Factor $15 x$ out of $15 x^{2} + 75 x$ .

Factor $15 x$ out of $15 x^{2}$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} - 5 x + 25) + 15 x (x) + 75 x} - 5$

Factor $15 x$ out of $75 x$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} - 5 x + 25) + 15 x (x) + 15 x (5)} - 5$

Factor $15 x$ out of $15 x (x) + 15 x (5)$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} - 5 x + 25) + 15 x (x + 5)} - 5$

Factor $x + 5$ out of $(x + 5) (x^{2} - 5 x + 25) + 15 x (x + 5)$ .

Factor $x + 5$ out of $15 x (x + 5)$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} - 5 x + 25) + (x + 5) (15 x)} - 5$

Factor $x + 5$ out of $(x + 5) (x^{2} - 5 x + 25) + (x + 5) (15 x)$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} - 5 x + 25 + 15 x)} - 5$

Add $- 5 x$ and $15 x$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} + 10 x + 25)} - 5$

Factor using the perfect square rule.

Rewrite $25$ as $5^{2}$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} + 10 x + 5^{2})} - 5$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Rewrite the polynomial.

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) (x^{2} + 2 \cdot x \cdot 5 + 5^{2})} - 5$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) {(x + 5)}^{2}} - 5$

Multiply $x + 5$ by ${(x + 5)}^{2}$ by adding the exponents.

Multiply $x + 5$ by ${(x + 5)}^{2}$ .

Raise $x + 5$ to the power of $1$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{(x + 5) {(x + 5)}^{2}} - 5$

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

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{{(x + 5)}^{1 + 2}} - 5$

Add $1$ and $2$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = \sqrt[3]{{(x + 5)}^{3}} - 5$

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

$f (x^{3} + 15 x^{2} + 75 x + 122) = x + 5 - 5$

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

Subtract $5$ from $5$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = x + 0$

Add $x$ and $0$ .

$f (x^{3} + 15 x^{2} + 75 x + 122) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} + 15 x^{2} + 75 x + 122$ is the inverse of $f (x) = \sqrt[3]{x + 3} - 5$ .

$f^{- 1} (x) = x^{3} + 15 x^{2} + 75 x + 122$

Find the Inverse cube root of x+3-5

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