Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Simplify the left side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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Step 3.3.1.1.1

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

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

Step 3.3.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

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

Step 3.3.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.2

Simplify.

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

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $1$ from both sides of the equation.

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

Step 3.4.2

Divide each term in $3 y = x^{3} - 1$ by $3$ and simplify.

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Step 3.4.2.1

Divide each term in $3 y = x^{3} - 1$ by $3$ .

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

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

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

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Combine the numerators over the common denominator.

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

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

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

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Step 5.2.4.1.1

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

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

Step 5.2.4.1.2

Cancel the common factor of $3$ .

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Step 5.2.4.1.2.1

Cancel the common factor.

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

Step 5.2.4.1.2.2

Rewrite the expression.

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

Step 5.2.4.2

Simplify.

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

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $3 x + 1 - 1$ .

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Step 5.2.5.1.1

Subtract $1$ from $1$ .

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

Step 5.2.5.1.2

Add $3 x$ and $0$ .

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

Step 5.2.5.2

Cancel the common factor of $3$ .

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Step 5.2.5.2.1

Cancel the common factor.

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

Step 5.2.5.2.2

Divide $x$ by $1$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.2

Rewrite the expression.

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

Step 5.3.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.3.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 5.3.3.3.2

Cancel the common factor.

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

Step 5.3.3.3.3

Rewrite the expression.

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

Step 5.3.4

Simplify by adding terms.

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Step 5.3.4.1

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

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Step 5.3.4.1.1

Add $- 1$ and $1$ .

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

Step 5.3.4.1.2

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

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

Step 5.3.4.2

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

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Step 5.3.4.2.1

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

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

Step 5.3.4.2.2

Cancel the common factor of $3$ .

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Step 5.3.4.2.2.1

Cancel the common factor.

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

Step 5.3.4.2.2.2

Rewrite the expression.

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

Step 5.4

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

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