Precalculus Examples

$f^{- 1} = \sqrt[3]{2 x - 9}$

Step 1

Rewrite the equation as $\sqrt[3]{2 x - 9} = f^{- 1}$ .

$\sqrt[3]{2 x - 9} = f^{- 1}$

Step 2

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

${\sqrt[3]{2 x - 9}}^{3} = {(f^{- 1})}^{3}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x - 9}$ as ${(2 x - 9)}^{\frac{1}{3}}$ .

${({(2 x - 9)}^{\frac{1}{3}})}^{3} = {(f^{- 1})}^{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(2 x - 9)}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(2 x - 9)}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.1.1.1

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

${(2 x - 9)}^{\frac{1}{3} \cdot 3} = {(f^{- 1})}^{3}$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(2 x - 9)}^{\frac{1}{3} \cdot 3} = {(f^{- 1})}^{3}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x - 9)}^{1} = {(f^{- 1})}^{3}$

Step 3.2.1.2

Simplify.

$2 x - 9 = {(f^{- 1})}^{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(f^{- 1})}^{3}$ .

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Step 3.3.1.1

Multiply the exponents in ${(f^{- 1})}^{3}$ .

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Step 3.3.1.1.1

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

$2 x - 9 = f^{- 1 \cdot 3}$

Step 3.3.1.1.2

Multiply $- 1$ by $3$ .

$2 x - 9 = f^{- 3}$

Step 3.3.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 x - 9 = \frac{1}{f^{3}}$

Step 4

Solve for $x$ .

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Step 4.1

Add $9$ to both sides of the equation.

$2 x = \frac{1}{f^{3}} + 9$

Step 4.2

Divide each term in $2 x = \frac{1}{f^{3}} + 9$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 x = \frac{1}{f^{3}} + 9$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{1}{f^{3}}}{2} + \frac{9}{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{1}{f^{3}}}{2} + \frac{9}{2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{1}{f^{3}}}{2} + \frac{9}{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1}{f^{3}} \cdot \frac{1}{2} + \frac{9}{2}$

Step 4.2.3.1.2

Combine.

$x = \frac{1 \cdot 1}{f^{3} \cdot 2} + \frac{9}{2}$

Step 4.2.3.1.3

Multiply $1$ by $1$ .

$x = \frac{1}{f^{3} \cdot 2} + \frac{9}{2}$

Step 4.2.3.1.4

Move $2$ to the left of $f^{3}$ .

$x = \frac{1}{2 f^{3}} + \frac{9}{2}$