Algebra Examples

$f (x) = - \frac{1}{2} {(x - 1)}^{3} + 4$

Step 1

Write $f (x) = - \frac{1}{2} {(x - 1)}^{3} + 4$ as an equation.

$y = - \frac{1}{2} {(x - 1)}^{3} + 4$

Step 2

Interchange the variables.

$x = - \frac{1}{2} \cdot {(y - 1)}^{3} + 4$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \frac{1}{2} \cdot {(y - 1)}^{3} + 4 = x$ .

$- \frac{1}{2} \cdot {(y - 1)}^{3} + 4 = x$

Step 3.2

Subtract $4$ from both sides of the equation.

$- \frac{1}{2} \cdot {(y - 1)}^{3} = x - 4$

Step 3.3

Combine ${(y - 1)}^{3}$ and $\frac{1}{2}$ .

$- \frac{{(y - 1)}^{3}}{2} = x - 4$

Step 3.4

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{{(y - 1)}^{3}}{2}) = - 2 (x - 4)$

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

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

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Step 3.5.1.1.1

Cancel the common factor of $2$ .

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Step 3.5.1.1.1.1

Move the leading negative in $- \frac{{(y - 1)}^{3}}{2}$ into the numerator.

$- 2 \frac{- {(y - 1)}^{3}}{2} = - 2 (x - 4)$

Step 3.5.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- {(y - 1)}^{3}}{2} = - 2 (x - 4)$

Step 3.5.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- {(y - 1)}^{3}}{2} = - 2 (x - 4)$

Step 3.5.1.1.1.4

Rewrite the expression.

$- - {(y - 1)}^{3} = - 2 (x - 4)$

Step 3.5.1.1.2

Multiply.

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Step 3.5.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 {(y - 1)}^{3} = - 2 (x - 4)$

Step 3.5.1.1.2.2

Multiply ${(y - 1)}^{3}$ by $1$ .

${(y - 1)}^{3} = - 2 (x - 4)$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $- 2 (x - 4)$ .

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Step 3.5.2.1.1

Apply the distributive property.

${(y - 1)}^{3} = - 2 x - 2 \cdot - 4$

Step 3.5.2.1.2

Multiply $- 2$ by $- 4$ .

${(y - 1)}^{3} = - 2 x + 8$

Step 3.6

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

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

Step 3.7

Simplify $\sqrt[3]{- 2 x + 8}$ .

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Step 3.7.1

Rewrite.

$y - 1 = 0 + 0 + \sqrt[3]{- 2 x + 8}$

Step 3.7.2

Simplify by adding zeros.

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

Step 3.7.3

Factor $2$ out of $- 2 x + 8$ .

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Step 3.7.3.1

Factor $2$ out of $- 2 x$ .

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

Step 3.7.3.2

Factor $2$ out of $8$ .

$y - 1 = \sqrt[3]{2 (- x) + 2 (4)}$

Step 3.7.3.3

Factor $2$ out of $2 (- x) + 2 (4)$ .

$y - 1 = \sqrt[3]{2 (- x + 4)}$

Step 3.8

Add $1$ to both sides of the equation.

$y = \sqrt[3]{2 (- x + 4)} + 1$

Step 4

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

$f^{- 1} (x) = \sqrt[3]{2 (- x + 4)} + 1$

Step 5

Verify if $f^{- 1} (x) = \sqrt[3]{2 (- x + 4)} + 1$ is the inverse of $f (x) = - \frac{1}{2} \cdot {(x - 1)}^{3} + 4$ .

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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 $f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) + 4)} + 1$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Use the Binomial Theorem.

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{1}{2} \cdot (x^{3} + 3 x^{2} \cdot - 1 + 3 x {(- 1)}^{2} + {(- 1)}^{3}) + 4) + 4)} + 1$

Step 5.2.3.1.2

Simplify each term.

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Step 5.2.3.1.2.1

Multiply $- 1$ by $3$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{1}{2} \cdot (x^{3} - 3 x^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) + 4) + 4)} + 1$

Step 5.2.3.1.2.2

Raise $- 1$ to the power of $2$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{1}{2} \cdot (x^{3} - 3 x^{2} + 3 x \cdot 1 + {(- 1)}^{3}) + 4) + 4)} + 1$

Step 5.2.3.1.2.3

Multiply $3$ by $1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{1}{2} \cdot (x^{3} - 3 x^{2} + 3 x + {(- 1)}^{3}) + 4) + 4)} + 1$

Step 5.2.3.1.2.4

Raise $- 1$ to the power of $3$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{1}{2} \cdot (x^{3} - 3 x^{2} + 3 x - 1) + 4) + 4)} + 1$

Step 5.2.3.1.3

Apply the distributive property.

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{1}{2} x^{3} - \frac{1}{2} \cdot (- 3 x^{2}) - \frac{1}{2} \cdot (3 x) - \frac{1}{2} \cdot - 1 + 4) + 4)} + 1$

Step 5.2.3.1.4

Simplify.

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Step 5.2.3.1.4.1

Combine $x^{3}$ and $\frac{1}{2}$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} - \frac{1}{2} \cdot (- 3 x^{2}) - \frac{1}{2} \cdot (3 x) - \frac{1}{2} \cdot - 1 + 4) + 4)} + 1$

Step 5.2.3.1.4.2

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

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Step 5.2.3.1.4.2.1

Multiply $- 3$ by $- 1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + 3 (\frac{1}{2}) \cdot x^{2} - \frac{1}{2} \cdot (3 x) - \frac{1}{2} \cdot - 1 + 4) + 4)} + 1$

Step 5.2.3.1.4.2.2

Combine $3$ and $\frac{1}{2}$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3}{2} x^{2} - \frac{1}{2} \cdot (3 x) - \frac{1}{2} \cdot - 1 + 4) + 4)} + 1$

Step 5.2.3.1.4.2.3

Combine $\frac{3}{2}$ and $x^{2}$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} - \frac{1}{2} \cdot (3 x) - \frac{1}{2} \cdot - 1 + 4) + 4)} + 1$

Step 5.2.3.1.4.3

Multiply $- \frac{1}{2} (3 x)$ .

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Step 5.2.3.1.4.3.1

Multiply $3$ by $- 1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} - 3 (\frac{1}{2}) \cdot x - \frac{1}{2} \cdot - 1 + 4) + 4)} + 1$

Step 5.2.3.1.4.3.2

Combine $- 3$ and $\frac{1}{2}$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} + \frac{- 3}{2} x - \frac{1}{2} \cdot - 1 + 4) + 4)} + 1$

Step 5.2.3.1.4.3.3

Combine $\frac{- 3}{2}$ and $x$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} + \frac{- 3 x}{2} - \frac{1}{2} \cdot - 1 + 4) + 4)} + 1$

Step 5.2.3.1.4.4

Multiply $- \frac{1}{2} \cdot - 1$ .

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Step 5.2.3.1.4.4.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} + \frac{- 3 x}{2} + 1 (\frac{1}{2}) + 4) + 4)} + 1$

Step 5.2.3.1.4.4.2

Multiply $\frac{1}{2}$ by $1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} + \frac{- 3 x}{2} + \frac{1}{2} + 4) + 4)} + 1$

Step 5.2.3.1.5

Move the negative in front of the fraction.

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} - \frac{3 x}{2} + \frac{1}{2} + 4) + 4)} + 1$

Step 5.2.3.2

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

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} - \frac{3 x}{2} + \frac{1}{2} + 4 \cdot \frac{2}{2}) + 4)} + 1$

Step 5.2.3.3

Combine $4$ and $\frac{2}{2}$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} - \frac{3 x}{2} + \frac{1}{2} + \frac{4 \cdot 2}{2}) + 4)} + 1$

Step 5.2.3.4

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} - \frac{3 x}{2} + \frac{1 + 4 \cdot 2}{2}) + 4)} + 1$

Step 5.2.3.5

Simplify the numerator.

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Step 5.2.3.5.1

Multiply $4$ by $2$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} - \frac{3 x}{2} + \frac{1 + 8}{2}) + 4)} + 1$

Step 5.2.3.5.2

Add $1$ and $8$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (- (- \frac{x^{3}}{2} + \frac{3 x^{2}}{2} - \frac{3 x}{2} + \frac{9}{2}) + 4)} + 1$

Step 5.2.3.6

Apply the distributive property.

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (\frac{x^{3}}{2} - \frac{3 x^{2}}{2} + \frac{3 x}{2} - \frac{9}{2} + 4)} + 1$

Step 5.2.3.7

Simplify.

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Step 5.2.3.7.1

Multiply $- - \frac{x^{3}}{2}$ .

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Step 5.2.3.7.1.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (1 (\frac{x^{3}}{2}) - \frac{3 x^{2}}{2} + \frac{3 x}{2} - \frac{9}{2} + 4)} + 1$

Step 5.2.3.7.1.2

Multiply $\frac{x^{3}}{2}$ by $1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (\frac{x^{3}}{2} - \frac{3 x^{2}}{2} + \frac{3 x}{2} - \frac{9}{2} + 4)} + 1$

Step 5.2.3.7.2

Multiply $- - \frac{3 x}{2}$ .

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Step 5.2.3.7.2.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (\frac{x^{3}}{2} - \frac{3 x^{2}}{2} + 1 (\frac{3 x}{2}) - \frac{9}{2} + 4)} + 1$

Step 5.2.3.7.2.2

Multiply $\frac{3 x}{2}$ by $1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (\frac{x^{3}}{2} - \frac{3 x^{2}}{2} + \frac{3 x}{2} - \frac{9}{2} + 4)} + 1$

Step 5.2.3.8

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

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (\frac{x^{3}}{2} - \frac{3 x^{2}}{2} + \frac{3 x}{2} - \frac{9}{2} + 4 \cdot \frac{2}{2})} + 1$

Step 5.2.3.9

Combine $4$ and $\frac{2}{2}$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (\frac{x^{3}}{2} - \frac{3 x^{2}}{2} + \frac{3 x}{2} - \frac{9}{2} + \frac{4 \cdot 2}{2})} + 1$

Step 5.2.3.10

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (\frac{x^{3}}{2} - \frac{3 x^{2}}{2} + \frac{3 x}{2} + \frac{- 9 + 4 \cdot 2}{2})} + 1$

Step 5.2.3.11

Simplify the numerator.

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Step 5.2.3.11.1

Multiply $4$ by $2$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (\frac{x^{3}}{2} - \frac{3 x^{2}}{2} + \frac{3 x}{2} + \frac{- 9 + 8}{2})} + 1$

Step 5.2.3.11.2

Add $- 9$ and $8$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (\frac{x^{3}}{2} - \frac{3 x^{2}}{2} + \frac{3 x}{2} + \frac{- 1}{2})} + 1$

Step 5.2.3.12

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{2 (\frac{x^{3} - 3 x^{2} + 3 x - 1}{2})} + 1$

Step 5.2.3.13

Combine $2$ and $\frac{x^{3} - 3 x^{2} + 3 x - 1}{2}$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{\frac{2 (x^{3} - 3 x^{2} + 3 x - 1)}{2}} + 1$

Step 5.2.3.14

Reduce the expression by cancelling the common factors.

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Step 5.2.3.14.1

Reduce the expression $\frac{2 (x^{3} - 3 x^{2} + 3 x - 1)}{2}$ by cancelling the common factors.

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Step 5.2.3.14.1.1

Cancel the common factor.

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{\frac{2 (x^{3} - 3 x^{2} + 3 x - 1)}{2}} + 1$

Step 5.2.3.14.1.2

Rewrite the expression.

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{\frac{x^{3} - 3 x^{2} + 3 x - 1}{1}} + 1$

Step 5.2.3.14.2

Divide $x^{3} - 3 x^{2} + 3 x - 1$ by $1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{x^{3} - 3 x^{2} + 3 x - 1} + 1$

Step 5.2.3.15

Factor $x^{3} - 3 x^{2} + 3 x - 1$ using the binomial theorem.

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = \sqrt[3]{{(x - 1)}^{3}} + 1$

Step 5.2.3.16

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

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = x - 1 + 1$

Step 5.2.4

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

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Step 5.2.4.1

Add $- 1$ and $1$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (- \frac{1}{2} \cdot {(x - 1)}^{3} + 4) = x$

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 $f (\sqrt[3]{2 (- x + 4)} + 1)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[3]{2 (- x + 4)} + 1) = - \frac{1}{2} \cdot {((\sqrt[3]{2 (- x + 4)} + 1) - 1)}^{3} + 4$

Step 5.3.3

Combine the opposite terms in $- \frac{1}{2} \cdot {((\sqrt[3]{2 (- x + 4)} + 1) - 1)}^{3} + 4$ .

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Step 5.3.3.1

Subtract $1$ from $1$ .

$f (\sqrt[3]{2 (- x + 4)} + 1) = - \frac{1}{2} \cdot {(\sqrt[3]{2 (- x + 4)} + 0)}^{3} + 4$

Step 5.3.3.2

Add $\sqrt[3]{2 (- x + 4)}$ and $0$ .

$f (\sqrt[3]{2 (- x + 4)} + 1) = - \frac{1}{2} \cdot {\sqrt[3]{2 (- x + 4)}}^{3} + 4$

Step 5.3.4

Simplify each term.

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Step 5.3.4.1

Rewrite ${\sqrt[3]{2 (- x + 4)}}^{3}$ as $2 (- x + 4)$ .

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Step 5.3.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 (- x + 4)}$ as ${(2 (- x + 4))}^{\frac{1}{3}}$ .

$f (\sqrt[3]{2 (- x + 4)} + 1) = - \frac{1}{2} \cdot {({(2 (- x + 4))}^{\frac{1}{3}})}^{3} + 4$

Step 5.3.4.1.2

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

$f (\sqrt[3]{2 (- x + 4)} + 1) = - \frac{1}{2} \cdot {(2 (- x + 4))}^{\frac{1}{3} \cdot 3} + 4$

Step 5.3.4.1.3

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

$f (\sqrt[3]{2 (- x + 4)} + 1) = - \frac{1}{2} \cdot {(2 (- x + 4))}^{\frac{3}{3}} + 4$

Step 5.3.4.1.4

Cancel the common factor of $3$ .

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Step 5.3.4.1.4.1

Cancel the common factor.

$f (\sqrt[3]{2 (- x + 4)} + 1) = - \frac{1}{2} \cdot {(2 (- x + 4))}^{\frac{3}{3}} + 4$

Step 5.3.4.1.4.2

Rewrite the expression.

$f (\sqrt[3]{2 (- x + 4)} + 1) = - \frac{1}{2} \cdot (2 (- x + 4)) + 4$

Step 5.3.4.1.5

Simplify.

$f (\sqrt[3]{2 (- x + 4)} + 1) = - \frac{1}{2} \cdot (2 (- x + 4)) + 4$

Step 5.3.4.2

Cancel the common factor of $2$ .

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Step 5.3.4.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (\sqrt[3]{2 (- x + 4)} + 1) = \frac{- 1}{2} \cdot (2 (- x + 4)) + 4$

Step 5.3.4.2.2

Cancel the common factor.

$f (\sqrt[3]{2 (- x + 4)} + 1) = \frac{- 1}{2} \cdot (2 (- x + 4)) + 4$

Step 5.3.4.2.3

Rewrite the expression.

$f (\sqrt[3]{2 (- x + 4)} + 1) = - 1 \cdot (- x + 4) + 4$

Step 5.3.4.3

Apply the distributive property.

$f (\sqrt[3]{2 (- x + 4)} + 1) = - 1 (- x) - 1 \cdot 4 + 4$

Step 5.3.4.4

Multiply $- 1 (- x)$ .

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Step 5.3.4.4.1

Multiply $- 1$ by $- 1$ .

$f (\sqrt[3]{2 (- x + 4)} + 1) = 1 x - 1 \cdot 4 + 4$

Step 5.3.4.4.2

Multiply $x$ by $1$ .

$f (\sqrt[3]{2 (- x + 4)} + 1) = x - 1 \cdot 4 + 4$

Step 5.3.4.5

Multiply $- 1$ by $4$ .

$f (\sqrt[3]{2 (- x + 4)} + 1) = x - 4 + 4$

Step 5.3.5

Combine the opposite terms in $x - 4 + 4$ .

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Step 5.3.5.1

Add $- 4$ and $4$ .

$f (\sqrt[3]{2 (- x + 4)} + 1) = x + 0$

Step 5.3.5.2

Add $x$ and $0$ .

$f (\sqrt[3]{2 (- x + 4)} + 1) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[3]{2 (- x + 4)} + 1$ is the inverse of $f (x) = - \frac{1}{2} \cdot {(x - 1)}^{3} + 4$ .

$f^{- 1} (x) = \sqrt[3]{2 (- x + 4)} + 1$