Find the Inverse f(x)=1/3 cube root of x-1

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

Step 1

Write $f (x) = \frac{1}{3} \sqrt[3]{x - 1}$ as an equation.

$y = \frac{1}{3} \sqrt[3]{x - 1}$

Step 2

Interchange the variables.

$x = \frac{1}{3} \cdot \sqrt[3]{y - 1}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{1}{3} \cdot \sqrt[3]{y - 1} = x$ .

$\frac{1}{3} \cdot \sqrt[3]{y - 1} = x$

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} \cdot \sqrt[3]{y - 1}) = 3 x$

Simplify the left side.

Simplify $3 (\frac{1}{3} \cdot \sqrt[3]{y - 1})$ .

Combine $\frac{1}{3}$ and $\sqrt[3]{y - 1}$ .

$3 \frac{\sqrt[3]{y - 1}}{3} = 3 x$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \frac{\sqrt[3]{y - 1}}{3} = 3 x$

Rewrite the expression.

$\sqrt[3]{y - 1} = 3 x$

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Apply the product rule to $3 x$ .

$y - 1 = 3^{3} x^{3}$

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

$y - 1 = 27 x^{3}$

Add $1$ to both sides of the equation.

$y = 27 x^{3} + 1$

Step 4

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

$f^{- 1} (x) = 27 x^{3} + 1$

Step 5

Verify if $f^{- 1} (x) = 27 x^{3} + 1$ is the inverse of $f (x) = \frac{1}{3} \cdot \sqrt[3]{x - 1}$ .

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

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 {(\frac{1}{3} \cdot \sqrt[3]{x - 1})}^{3} + 1$

Simplify each term.

Combine $\frac{1}{3}$ and $\sqrt[3]{x - 1}$ .

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 {(\frac{\sqrt[3]{x - 1}}{3})}^{3} + 1$

Apply the product rule to $\frac{\sqrt[3]{x - 1}}{3}$ .

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 (\frac{{\sqrt[3]{x - 1}}^{3}}{3^{3}}) + 1$

Rewrite ${\sqrt[3]{x - 1}}^{3}$ as $x - 1$ .

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

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 (\frac{{({(x - 1)}^{\frac{1}{3}})}^{3}}{3^{3}}) + 1$

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

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 (\frac{{(x - 1)}^{\frac{1}{3} \cdot 3}}{3^{3}}) + 1$

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

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 (\frac{{(x - 1)}^{\frac{3}{3}}}{3^{3}}) + 1$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 (\frac{{(x - 1)}^{\frac{3}{3}}}{3^{3}}) + 1$

Rewrite the expression.

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 (\frac{x - 1}{3^{3}}) + 1$

Simplify.

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 (\frac{x - 1}{3^{3}}) + 1$

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

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 (\frac{x - 1}{27}) + 1$

Cancel the common factor of $27$ .

Cancel the common factor.

$f^{- 1} (\frac{1}{3} \cdot \sqrt[3]{x - 1}) = 27 (\frac{x - 1}{27}) + 1$

Rewrite the expression.

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

Combine the opposite terms in $x - 1 + 1$ .

Add $- 1$ and $1$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (27 x^{3} + 1)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (27 x^{3} + 1) = \frac{1}{3} \cdot \sqrt[3]{(27 x^{3} + 1) - 1}$

Simplify by subtracting numbers.

Subtract $1$ from $1$ .

$f (27 x^{3} + 1) = \frac{1}{3} \cdot \sqrt[3]{27 x^{3} + 0}$

Add $27 x^{3}$ and $0$ .

$f (27 x^{3} + 1) = \frac{1}{3} \cdot \sqrt[3]{27 x^{3}}$

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

$f (27 x^{3} + 1) = \frac{1}{3} \cdot \sqrt[3]{{(3 x)}^{3}}$

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

$f (27 x^{3} + 1) = \frac{1}{3} \cdot (3 x)$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$f (27 x^{3} + 1) = \frac{1}{3} \cdot (3 (x))$

Cancel the common factor.

$f (27 x^{3} + 1) = \frac{1}{3} \cdot (3 x)$

Rewrite the expression.

$f (27 x^{3} + 1) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 27 x^{3} + 1$ is the inverse of $f (x) = \frac{1}{3} \cdot \sqrt[3]{x - 1}$ .

$f^{- 1} (x) = 27 x^{3} + 1$

Find the Inverse f(x)=1/3 cube root of x-1

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