Precalculus Examples

$f (x) = \frac{3 - 8 x^{3}}{2}$

Step 1

Write $f (x) = \frac{3 - 8 x^{3}}{2}$ as an equation.

$y = \frac{3 - 8 x^{3}}{2}$

Step 2

Interchange the variables.

$x = \frac{3 - 8 y^{3}}{2}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{3 - 8 y^{3}}{2} = x$ .

$\frac{3 - 8 y^{3}}{2} = x$

Step 3.2

Multiply both sides by $2$ .

$\frac{3 - 8 y^{3}}{2} \cdot 2 = x \cdot 2$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{3 - 8 y^{3}}{2} \cdot 2$ .

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Step 3.3.1.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{3 - 8 y^{3}}{2} \cdot 2 = x \cdot 2$

Step 3.3.1.1.1.2

Rewrite the expression.

$3 - 8 y^{3} = x \cdot 2$

Step 3.3.1.1.2

Reorder $3$ and $- 8 y^{3}$ .

$- 8 y^{3} + 3 = x \cdot 2$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Move $2$ to the left of $x$ .

$- 8 y^{3} + 3 = 2 x$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $3$ from both sides of the equation.

$- 8 y^{3} = 2 x - 3$

Step 3.4.2

Divide each term in $- 8 y^{3} = 2 x - 3$ by $- 8$ and simplify.

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Step 3.4.2.1

Divide each term in $- 8 y^{3} = 2 x - 3$ by $- 8$ .

$\frac{- 8 y^{3}}{- 8} = \frac{2 x}{- 8} + \frac{- 3}{- 8}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $- 8$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{- 8 y^{3}}{- 8} = \frac{2 x}{- 8} + \frac{- 3}{- 8}$

Step 3.4.2.2.1.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{2 x}{- 8} + \frac{- 3}{- 8}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Cancel the common factor of $2$ and $- 8$ .

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Step 3.4.2.3.1.1.1

Factor $2$ out of $2 x$ .

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

Step 3.4.2.3.1.1.2

Cancel the common factors.

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Step 3.4.2.3.1.1.2.1

Factor $2$ out of $- 8$ .

$y^{3} = \frac{2 x}{2 \cdot - 4} + \frac{- 3}{- 8}$

Step 3.4.2.3.1.1.2.2

Cancel the common factor.

$y^{3} = \frac{2 x}{2 \cdot - 4} + \frac{- 3}{- 8}$

Step 3.4.2.3.1.1.2.3

Rewrite the expression.

$y^{3} = \frac{x}{- 4} + \frac{- 3}{- 8}$

Step 3.4.2.3.1.2

Move the negative in front of the fraction.

$y^{3} = - \frac{x}{4} + \frac{- 3}{- 8}$

Step 3.4.2.3.1.3

Dividing two negative values results in a positive value.

$y^{3} = - \frac{x}{4} + \frac{3}{8}$

Step 3.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{- \frac{x}{4} + \frac{3}{8}}$

Step 3.4.4

Simplify $\sqrt[3]{- \frac{x}{4} + \frac{3}{8}}$ .

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Step 3.4.4.1

To write $- \frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \sqrt[3]{- \frac{x}{4} \cdot \frac{2}{2} + \frac{3}{8}}$

Step 3.4.4.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.4.2.1

Multiply $\frac{x}{4}$ by $\frac{2}{2}$ .

$y = \sqrt[3]{- \frac{x \cdot 2}{4 \cdot 2} + \frac{3}{8}}$

Step 3.4.4.2.2

Multiply $4$ by $2$ .

$y = \sqrt[3]{- \frac{x \cdot 2}{8} + \frac{3}{8}}$

Step 3.4.4.3

Combine the numerators over the common denominator.

$y = \sqrt[3]{\frac{- x \cdot 2 + 3}{8}}$

Step 3.4.4.4

Multiply $2$ by $- 1$ .

$y = \sqrt[3]{\frac{- 2 x + 3}{8}}$

Step 3.4.4.5

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

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Step 3.4.4.5.1

Factor the perfect power $1^{3}$ out of $- 2 x + 3$ .

$y = \sqrt[3]{\frac{1^{3} (- 2 x + 3)}{8}}$

Step 3.4.4.5.2

Factor the perfect power $2^{3}$ out of $8$ .

$y = \sqrt[3]{\frac{1^{3} (- 2 x + 3)}{2^{3} \cdot 1}}$

Step 3.4.4.5.3

Rearrange the fraction $\frac{1^{3} (- 2 x + 3)}{2^{3} \cdot 1}$ .

$y = \sqrt[3]{{(\frac{1}{2})}^{3} (- 2 x + 3)}$

Step 3.4.4.6

Pull terms out from under the radical.

$y = \frac{1}{2} \sqrt[3]{- 2 x + 3}$

Step 3.4.4.7

Combine $\frac{1}{2}$ and $\sqrt[3]{- 2 x + 3}$ .

$y = \frac{\sqrt[3]{- 2 x + 3}}{2}$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[3]{- 2 x + 3}}{2}$ is the inverse of $f (x) = \frac{3 - 8 x^{3}}{2}$ .

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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 $f^{- 1} (\frac{3 - 8 x^{3}}{2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{\sqrt[3]{- 2 \frac{3 - 8 x^{3}}{2} + 3}}{2}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Cancel the common factor of $2$ .

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Step 5.2.3.1.1

Factor $2$ out of $- 2$ .

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{\sqrt[3]{2 (- 1) (\frac{3 - 8 x^{3}}{2}) + 3}}{2}$

Step 5.2.3.1.2

Cancel the common factor.

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{\sqrt[3]{2 \cdot (- 1 \frac{3 - 8 x^{3}}{2}) + 3}}{2}$

Step 5.2.3.1.3

Rewrite the expression.

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{\sqrt[3]{- 1 (3 - 8 x^{3}) + 3}}{2}$

Step 5.2.3.2

Apply the distributive property.

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{\sqrt[3]{- 1 \cdot 3 - 1 (- 8 x^{3}) + 3}}{2}$

Step 5.2.3.3

Multiply $- 1$ by $3$ .

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{\sqrt[3]{- 3 - 1 (- 8 x^{3}) + 3}}{2}$

Step 5.2.3.4

Multiply $- 8$ by $- 1$ .

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{\sqrt[3]{- 3 + 8 x^{3} + 3}}{2}$

Step 5.2.3.5

Add $- 3$ and $3$ .

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{\sqrt[3]{8 x^{3} + 0}}{2}$

Step 5.2.3.6

Add $8 x^{3}$ and $0$ .

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{\sqrt[3]{8 x^{3}}}{2}$

Step 5.2.3.7

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{\sqrt[3]{{(2 x)}^{3}}}{2}$

Step 5.2.3.8

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

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{2 x}{2}$

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = \frac{2 x}{2}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{3 - 8 x^{3}}{2}) = x$

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 $f (\frac{\sqrt[3]{- 2 x + 3}}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 - 8 {(\frac{\sqrt[3]{- 2 x + 3}}{2})}^{3}}{2}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

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

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 - 8 {(\frac{{(- 2 x + 3)}^{\frac{1}{3}}}{2})}^{3}}{2}$

Step 5.3.3.2

Apply the product rule to $\frac{{(- 2 x + 3)}^{\frac{1}{3}}}{2}$ .

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 - 8 \frac{{({(- 2 x + 3)}^{\frac{1}{3}})}^{3}}{2^{3}}}{2}$

Step 5.3.3.3

Simplify the numerator.

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Step 5.3.3.3.1

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

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Step 5.3.3.3.1.1

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

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 - 8 \frac{{(- 2 x + 3)}^{\frac{1}{3} \cdot 3}}{2^{3}}}{2}$

Step 5.3.3.3.1.2

Cancel the common factor of $3$ .

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Step 5.3.3.3.1.2.1

Cancel the common factor.

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 - 8 \frac{{(- 2 x + 3)}^{\frac{1}{3} \cdot 3}}{2^{3}}}{2}$

Step 5.3.3.3.1.2.2

Rewrite the expression.

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 - 8 \frac{- 2 x + 3}{2^{3}}}{2}$

Step 5.3.3.3.2

Simplify.

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 - 8 \frac{- 2 x + 3}{2^{3}}}{2}$

Step 5.3.3.4

Raise $2$ to the power of $3$ .

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 - 8 \frac{- 2 x + 3}{8}}{2}$

Step 5.3.3.5

Cancel the common factor of $8$ .

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Step 5.3.3.5.1

Factor $8$ out of $- 8$ .

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 + 8 (- 1) (\frac{- 2 x + 3}{8})}{2}$

Step 5.3.3.5.2

Cancel the common factor.

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 + 8 \cdot (- 1 \frac{- 2 x + 3}{8})}{2}$

Step 5.3.3.5.3

Rewrite the expression.

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

Step 5.3.3.6

Apply the distributive property.

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 - 1 (- 2 x) - 1 \cdot 3}{2}$

Step 5.3.3.7

Multiply $- 2$ by $- 1$ .

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 + 2 x - 1 \cdot 3}{2}$

Step 5.3.3.8

Multiply $- 1$ by $3$ .

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{3 + 2 x - 3}{2}$

Step 5.3.3.9

Subtract $3$ from $3$ .

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{2 x + 0}{2}$

Step 5.3.3.10

Add $2 x$ and $0$ .

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{2 x}{2}$

Step 5.3.4

Cancel the common factor of $2$ .

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Step 5.3.4.1

Cancel the common factor.

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = \frac{2 x}{2}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (\frac{\sqrt[3]{- 2 x + 3}}{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[3]{- 2 x + 3}}{2}$ is the inverse of $f (x) = \frac{3 - 8 x^{3}}{2}$ .

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