Find the Inverse f(x)=5(x^(1/3)-8)

$f (x) = 5 (x^{\frac{1}{3}} - 8)$

Write $f (x) = 5 (x^{\frac{1}{3}} - 8)$ as an equation.

$y = 5 (x^{\frac{1}{3}} - 8)$

Interchange the variables.

$x = 5 (y^{\frac{1}{3}} - 8)$

Solve for $y$ .

Rewrite the equation as $5 (y^{\frac{1}{3}} - 8) = x$ .

$5 (y^{\frac{1}{3}} - 8) = x$

Divide each term in $5 (y^{\frac{1}{3}} - 8) = x$ by $5$ and simplify.

Divide each term in $5 (y^{\frac{1}{3}} - 8) = x$ by $5$ .

$\frac{5 (y^{\frac{1}{3}} - 8)}{5} = \frac{x}{5}$

Simplify the left side.

Cancel the common factor.

$\frac{5 (y^{\frac{1}{3}} - 8)}{5} = \frac{x}{5}$

Divide $y^{\frac{1}{3}} - 8$ by $1$ .

$y^{\frac{1}{3}} - 8 = \frac{x}{5}$

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

Add $8$ to both sides of the equation.

$y^{\frac{1}{3}} - \frac{x}{5} = 8$

Add $\frac{x}{5}$ to both sides of the equation.

$y^{\frac{1}{3}} = 8 + \frac{x}{5}$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{3}})}^{3} = {(8 + \frac{x}{5})}^{3}$

Simplify the exponent.

Simplify the left side.

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

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

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

$y^{\frac{1}{3} \cdot 3} = {(8 + \frac{x}{5})}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(8 + \frac{x}{5})}^{3}$

Rewrite the expression.

$y^{1} = {(8 + \frac{x}{5})}^{3}$

Simplify.

$y = {(8 + \frac{x}{5})}^{3}$

Simplify the right side.

Simplify ${(8 + \frac{x}{5})}^{3}$ .

Use the Binomial Theorem.

$y = 8^{3} + 3 \cdot 8^{2} \frac{x}{5} + 3 \cdot 8 {(\frac{x}{5})}^{2} + {(\frac{x}{5})}^{3}$

Simplify each term.

Raise $8$ to the power of $3$ .

$y = 512 + 3 \cdot 8^{2} \frac{x}{5} + 3 \cdot 8 {(\frac{x}{5})}^{2} + {(\frac{x}{5})}^{3}$

Raise $8$ to the power of $2$ .

$y = 512 + 3 \cdot 64 \frac{x}{5} + 3 \cdot 8 {(\frac{x}{5})}^{2} + {(\frac{x}{5})}^{3}$

Multiply $3$ by $64$ .

$y = 512 + 192 \frac{x}{5} + 3 \cdot 8 {(\frac{x}{5})}^{2} + {(\frac{x}{5})}^{3}$

Combine $192$ and $\frac{x}{5}$ .

$y = 512 + \frac{192 x}{5} + 3 \cdot 8 {(\frac{x}{5})}^{2} + {(\frac{x}{5})}^{3}$

Multiply $3$ by $8$ .

$y = 512 + \frac{192 x}{5} + 24 {(\frac{x}{5})}^{2} + {(\frac{x}{5})}^{3}$

Apply the product rule to $\frac{x}{5}$ .

$y = 512 + \frac{192 x}{5} + 24 \frac{x^{2}}{5^{2}} + {(\frac{x}{5})}^{3}$

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

$y = 512 + \frac{192 x}{5} + 24 \frac{x^{2}}{25} + {(\frac{x}{5})}^{3}$

Combine $24$ and $\frac{x^{2}}{25}$ .

$y = 512 + \frac{192 x}{5} + \frac{24 x^{2}}{25} + {(\frac{x}{5})}^{3}$

Apply the product rule to $\frac{x}{5}$ .

$y = 512 + \frac{192 x}{5} + \frac{24 x^{2}}{25} + \frac{x^{3}}{5^{3}}$

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

$y = 512 + \frac{192 x}{5} + \frac{24 x^{2}}{25} + \frac{x^{3}}{125}$

Simplify $512 + \frac{192 x}{5} + \frac{24 x^{2}}{25} + \frac{x^{3}}{125}$ .

Move $512$ .

$y = \frac{192 x}{5} + \frac{24 x^{2}}{25} + \frac{x^{3}}{125} + 512$

Move $\frac{192 x}{5}$ .

$y = \frac{24 x^{2}}{25} + \frac{x^{3}}{125} + \frac{192 x}{5} + 512$

Reorder $\frac{24 x^{2}}{25}$ and $\frac{x^{3}}{125}$ .

$y = \frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512$

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

$f^{- 1} (x) = \frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512$

Verify if $f^{- 1} (x) = \frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512$ is the inverse of $f (x) = 5 (x^{\frac{1}{3}} - 8)$ .

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 $f^{- 1} (5 (x^{\frac{1}{3}} - 8))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = \frac{{(5 (x^{\frac{1}{3}} - 8))}^{3}}{125} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Simplify each term.

Simplify the numerator.

Apply the product rule to $5 (x^{\frac{1}{3}} - 8)$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = \frac{5^{3} {(x^{\frac{1}{3}} - 8)}^{3}}{125} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

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

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = \frac{125 {(x^{\frac{1}{3}} - 8)}^{3}}{125} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Cancel the common factor.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = \frac{125 {(x^{\frac{1}{3}} - 8)}^{3}}{125} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Divide ${(x^{\frac{1}{3}} - 8)}^{3}$ by $1$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = {(x^{\frac{1}{3}} - 8)}^{3} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Use the Binomial Theorem.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = {(x^{\frac{1}{3}})}^{3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot - 8 + 3 x^{\frac{1}{3}} {(- 8)}^{2} + {(- 8)}^{3} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Simplify each term.

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

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

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x^{\frac{1}{3} \cdot 3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot - 8 + 3 x^{\frac{1}{3}} {(- 8)}^{2} + {(- 8)}^{3} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Cancel the common factor of $3$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x + 3 {(x^{\frac{1}{3}})}^{2} \cdot - 8 + 3 x^{\frac{1}{3}} {(- 8)}^{2} + {(- 8)}^{3} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Simplify.

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

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

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x + 3 x^{\frac{1}{3} \cdot 2} \cdot - 8 + 3 x^{\frac{1}{3}} {(- 8)}^{2} + {(- 8)}^{3} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Combine $\frac{1}{3}$ and $2$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x + 3 x^{\frac{2}{3}} \cdot - 8 + 3 x^{\frac{1}{3}} {(- 8)}^{2} + {(- 8)}^{3} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Multiply $- 8$ by $3$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} {(- 8)}^{2} + {(- 8)}^{3} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Raise $- 8$ to the power of $2$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 64 + {(- 8)}^{3} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Multiply $64$ by $3$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} + {(- 8)}^{3} + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Raise $- 8$ to the power of $3$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + \frac{24 {(5 (x^{\frac{1}{3}} - 8))}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Simplify the numerator.

Apply the product rule to $5 (x^{\frac{1}{3}} - 8)$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + \frac{24 \cdot (5^{2} {(x^{\frac{1}{3}} - 8)}^{2})}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

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

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + \frac{24 \cdot (25 {(x^{\frac{1}{3}} - 8)}^{2})}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Multiply $24$ by $25$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + \frac{600 {(x^{\frac{1}{3}} - 8)}^{2}}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Factor $25$ out of $600 {(x^{\frac{1}{3}} - 8)}^{2}$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + \frac{25 (24 {(x^{\frac{1}{3}} - 8)}^{2})}{25} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Cancel the common factors.

Factor $25$ out of $25$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + \frac{25 (24 {(x^{\frac{1}{3}} - 8)}^{2})}{25 (1)} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Cancel the common factor.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + \frac{25 (24 {(x^{\frac{1}{3}} - 8)}^{2})}{25 \cdot 1} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Rewrite the expression.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + \frac{24 {(x^{\frac{1}{3}} - 8)}^{2}}{1} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Divide $24 {(x^{\frac{1}{3}} - 8)}^{2}$ by $1$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 {(x^{\frac{1}{3}} - 8)}^{2} + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Rewrite ${(x^{\frac{1}{3}} - 8)}^{2}$ as $(x^{\frac{1}{3}} - 8) (x^{\frac{1}{3}} - 8)$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 ((x^{\frac{1}{3}} - 8) (x^{\frac{1}{3}} - 8)) + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Expand $(x^{\frac{1}{3}} - 8) (x^{\frac{1}{3}} - 8)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 (x^{\frac{1}{3}} (x^{\frac{1}{3}} - 8) - 8 (x^{\frac{1}{3}} - 8)) + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Apply the distributive property.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot - 8 - 8 (x^{\frac{1}{3}} - 8)) + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Apply the distributive property.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot - 8 - 8 x^{\frac{1}{3}} - 8 \cdot - 8) + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Simplify and combine like terms.

Simplify each term.

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 (x^{\frac{1}{3} + \frac{1}{3}} + x^{\frac{1}{3}} \cdot - 8 - 8 x^{\frac{1}{3}} - 8 \cdot - 8) + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Combine the numerators over the common denominator.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 (x^{\frac{1 + 1}{3}} + x^{\frac{1}{3}} \cdot - 8 - 8 x^{\frac{1}{3}} - 8 \cdot - 8) + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Add $1$ and $1$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 (x^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot - 8 - 8 x^{\frac{1}{3}} - 8 \cdot - 8) + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Move $- 8$ to the left of $x^{\frac{1}{3}}$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 (x^{\frac{2}{3}} - 8 \cdot x^{\frac{1}{3}} - 8 x^{\frac{1}{3}} - 8 \cdot - 8) + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Multiply $- 8$ by $- 8$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 (x^{\frac{2}{3}} - 8 x^{\frac{1}{3}} - 8 x^{\frac{1}{3}} + 64) + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Subtract $8 x^{\frac{1}{3}}$ from $- 8 x^{\frac{1}{3}}$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 (x^{\frac{2}{3}} - 16 x^{\frac{1}{3}} + 64) + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Apply the distributive property.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 x^{\frac{2}{3}} + 24 (- 16 x^{\frac{1}{3}}) + 24 \cdot 64 + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Simplify.

Multiply $- 16$ by $24$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 x^{\frac{2}{3}} - 384 x^{\frac{1}{3}} + 24 \cdot 64 + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Multiply $24$ by $64$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 x^{\frac{2}{3}} - 384 x^{\frac{1}{3}} + 1536 + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Cancel the common factor.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 x^{\frac{2}{3}} - 384 x^{\frac{1}{3}} + 1536 + \frac{192 (5 (x^{\frac{1}{3}} - 8))}{5} + 512$

Divide $192 (x^{\frac{1}{3}} - 8)$ by $1$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 x^{\frac{2}{3}} - 384 x^{\frac{1}{3}} + 1536 + 192 (x^{\frac{1}{3}} - 8) + 512$

Apply the distributive property.

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 x^{\frac{2}{3}} - 384 x^{\frac{1}{3}} + 1536 + 192 x^{\frac{1}{3}} + 192 \cdot - 8 + 512$

Multiply $192$ by $- 8$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 x^{\frac{2}{3}} - 384 x^{\frac{1}{3}} + 1536 + 192 x^{\frac{1}{3}} - 1536 + 512$

Simplify by adding terms.

Combine the opposite terms in $x - 24 x^{\frac{2}{3}} + 192 x^{\frac{1}{3}} - 512 + 24 x^{\frac{2}{3}} - 384 x^{\frac{1}{3}} + 1536 + 192 x^{\frac{1}{3}} - 1536 + 512$ .

Add $- 24 x^{\frac{2}{3}}$ and $24 x^{\frac{2}{3}}$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x + 192 x^{\frac{1}{3}} - 512 + 0 - 384 x^{\frac{1}{3}} + 1536 + 192 x^{\frac{1}{3}} - 1536 + 512$

Add $x + 192 x^{\frac{1}{3}} - 512$ and $0$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x + 192 x^{\frac{1}{3}} - 512 - 384 x^{\frac{1}{3}} + 1536 + 192 x^{\frac{1}{3}} - 1536 + 512$

Subtract $1536$ from $1536$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x + 192 x^{\frac{1}{3}} - 512 - 384 x^{\frac{1}{3}} + 192 x^{\frac{1}{3}} + 0 + 512$

Add $x + 192 x^{\frac{1}{3}} - 512 - 384 x^{\frac{1}{3}} + 192 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x + 192 x^{\frac{1}{3}} - 512 - 384 x^{\frac{1}{3}} + 192 x^{\frac{1}{3}} + 512$

Add $- 512$ and $512$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x + 192 x^{\frac{1}{3}} - 384 x^{\frac{1}{3}} + 192 x^{\frac{1}{3}} + 0$

Add $x + 192 x^{\frac{1}{3}} - 384 x^{\frac{1}{3}} + 192 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x + 192 x^{\frac{1}{3}} - 384 x^{\frac{1}{3}} + 192 x^{\frac{1}{3}}$

Subtract $384 x^{\frac{1}{3}}$ from $192 x^{\frac{1}{3}}$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x - 192 x^{\frac{1}{3}} + 192 x^{\frac{1}{3}}$

Combine the opposite terms in $x - 192 x^{\frac{1}{3}} + 192 x^{\frac{1}{3}}$ .

Add $- 192 x^{\frac{1}{3}}$ and $192 x^{\frac{1}{3}}$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (5 (x^{\frac{1}{3}} - 8)) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512) = 5 ({(\frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512)}^{\frac{1}{3}} - 8)$

Apply the distributive property.

$f (\frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512) = 5 {(\frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512)}^{\frac{1}{3}} + 5 \cdot - 8$

Multiply $5$ by $- 8$ .

$f (\frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512) = 5 {(\frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512)}^{\frac{1}{3}} - 40$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512$ is the inverse of $f (x) = 5 (x^{\frac{1}{3}} - 8)$ .

$f^{- 1} (x) = \frac{x^{3}}{125} + \frac{24 x^{2}}{25} + \frac{192 x}{5} + 512$

Find the Inverse f(x)=5(x^(1/3)-8)

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