Find the Inverse f(x)=8 cube root of x-3

$f (x) = 8 \sqrt[3]{x - 3}$

Write $f (x) = 8 \sqrt[3]{x - 3}$ as an equation.

$y = 8 \sqrt[3]{x - 3}$

Interchange the variables.

$x = 8 \sqrt[3]{y - 3}$

Solve for $y$ .

Rewrite the equation as $8 \sqrt[3]{y - 3} = x$ .

$8 \sqrt[3]{y - 3} = x$

Divide each term in $8 \sqrt[3]{y - 3} = x$ by $8$ and simplify.

Divide each term in $8 \sqrt[3]{y - 3} = x$ by $8$ .

$\frac{8 \sqrt[3]{y - 3}}{8} = \frac{x}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 \sqrt[3]{y - 3}}{8} = \frac{x}{8}$

Divide $\sqrt[3]{y - 3}$ by $1$ .

$\sqrt[3]{y - 3} = \frac{x}{8}$

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

${\sqrt[3]{y - 3}}^{3} = {(\frac{x}{8})}^{3}$

Simplify each side of the equation.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Apply the product rule to $\frac{x}{8}$ .

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

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

$y - 3 = \frac{x^{3}}{512}$

Add $3$ to both sides of the equation.

$y = \frac{x^{3}}{512} + 3$

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

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

Verify if $f^{- 1} (x) = \frac{x^{3}}{512} + 3$ is the inverse of $f (x) = 8 \sqrt[3]{x - 3}$ .

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

$f^{- 1} (8 \sqrt[3]{x - 3}) = \frac{{(8 \sqrt[3]{x - 3})}^{3}}{512} + 3$

Simplify each term.

Simplify the numerator.

Apply the product rule to $8 \sqrt[3]{x - 3}$ .

$f^{- 1} (8 \sqrt[3]{x - 3}) = \frac{8^{3} {\sqrt[3]{x - 3}}^{3}}{512} + 3$

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

$f^{- 1} (8 \sqrt[3]{x - 3}) = \frac{512 {\sqrt[3]{x - 3}}^{3}}{512} + 3$

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

$f^{- 1} (8 \sqrt[3]{x - 3}) = \frac{512 {({(x - 3)}^{\frac{1}{3}})}^{3}}{512} + 3$

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

$f^{- 1} (8 \sqrt[3]{x - 3}) = \frac{512 {(x - 3)}^{\frac{1}{3} \cdot 3}}{512} + 3$

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

$f^{- 1} (8 \sqrt[3]{x - 3}) = \frac{512 {(x - 3)}^{\frac{3}{3}}}{512} + 3$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (8 \sqrt[3]{x - 3}) = \frac{512 {(x - 3)}^{\frac{3}{3}}}{512} + 3$

Rewrite the expression.

$f^{- 1} (8 \sqrt[3]{x - 3}) = \frac{512 (x - 3)}{512} + 3$

Simplify.

$f^{- 1} (8 \sqrt[3]{x - 3}) = \frac{512 (x - 3)}{512} + 3$

Cancel the common factor of $512$ .

Cancel the common factor.

$f^{- 1} (8 \sqrt[3]{x - 3}) = \frac{512 (x - 3)}{512} + 3$

Divide $x - 3$ by $1$ .

$f^{- 1} (8 \sqrt[3]{x - 3}) = x - 3 + 3$

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

Add $- 3$ and $3$ .

$f^{- 1} (8 \sqrt[3]{x - 3}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (8 \sqrt[3]{x - 3}) = x$

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

Set up the composite result function.

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

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

$f (\frac{x^{3}}{512} + 3) = 8 \sqrt[3]{(\frac{x^{3}}{512} + 3) - 3}$

Subtract $3$ from $3$ .

$f (\frac{x^{3}}{512} + 3) = 8 \sqrt[3]{\frac{x^{3}}{512} + 0}$

Add $\frac{x^{3}}{512}$ and $0$ .

$f (\frac{x^{3}}{512} + 3) = 8 \sqrt[3]{\frac{x^{3}}{512}}$

Rewrite $512$ as $8^{3}$ .

$f (\frac{x^{3}}{512} + 3) = 8 \sqrt[3]{\frac{x^{3}}{8^{3}}}$

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

$f (\frac{x^{3}}{512} + 3) = 8 \sqrt[3]{{(\frac{x}{8})}^{3}}$

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

$f (\frac{x^{3}}{512} + 3) = 8 (\frac{x}{8})$

Cancel the common factor of $8$ .

Cancel the common factor.

$f (\frac{x^{3}}{512} + 3) = 8 (\frac{x}{8})$

Rewrite the expression.

$f (\frac{x^{3}}{512} + 3) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{512} + 3$ is the inverse of $f (x) = 8 \sqrt[3]{x - 3}$ .

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

Find the Inverse f(x)=8 cube root of x-3

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