Precalculus Examples

$f (x) = 2 (x^{7} + 2)$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = 2 (x^{7} + 2)$ as an equation.

$y = 2 (x^{7} + 2)$

Step 2

Interchange the variables.

$x = 2 (y^{7} + 2)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 (y^{7} + 2) = x$ .

$2 (y^{7} + 2) = x$

Step 3.2

Divide each term in $2 (y^{7} + 2) = x$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 (y^{7} + 2) = x$ by $2$ .

$\frac{2 (y^{7} + 2)}{2} = \frac{x}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 (y^{7} + 2)}{2} = \frac{x}{2}$

Step 3.2.2.1.2

Divide $y^{7} + 2$ by $1$ .

$y^{7} + 2 = \frac{x}{2}$

Step 3.3

Subtract $2$ from both sides of the equation.

$y^{7} = \frac{x}{2} - 2$

Step 3.4

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

$y = \sqrt[7]{\frac{x}{2} - 2}$

Step 3.5

Simplify $\sqrt[7]{\frac{x}{2} - 2}$ .

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Step 3.5.1

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

$y = \sqrt[7]{\frac{x}{2} - 2 \cdot \frac{2}{2}}$

Step 3.5.2

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

$y = \sqrt[7]{\frac{x}{2} + \frac{- 2 \cdot 2}{2}}$

Step 3.5.3

Combine the numerators over the common denominator.

$y = \sqrt[7]{\frac{x - 2 \cdot 2}{2}}$

Step 3.5.4

Multiply $- 2$ by $2$ .

$y = \sqrt[7]{\frac{x - 4}{2}}$

Step 3.5.5

Rewrite $\sqrt[7]{\frac{x - 4}{2}}$ as $\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}$ .

$y = \frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}$ is the inverse of $f (x) = 2 (x^{7} + 2)$ .

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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 $f^{- 1} (2 (x^{7} + 2))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 (x^{7} + 2)) = \frac{\sqrt[7]{(2 (x^{7} + 2)) - 4}}{\sqrt[7]{2}}$

Step 5.2.3

Combine $\sqrt[7]{2 (x^{7} + 2) - 4}$ and $\sqrt[7]{2}$ into a single radical.

$f^{- 1} (2 (x^{7} + 2)) = \sqrt[7]{\frac{2 (x^{7} + 2) - 4}{2}}$

Step 5.2.4

Factor $2$ out of $2 (x^{7} + 2) - 4$ .

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Step 5.2.4.1

Factor $2$ out of $- 4$ .

$f^{- 1} (2 (x^{7} + 2)) = \sqrt[7]{\frac{2 (x^{7} + 2) + 2 \cdot - 2}{2}}$

Step 5.2.4.2

Factor $2$ out of $2 (x^{7} + 2) + 2 \cdot - 2$ .

$f^{- 1} (2 (x^{7} + 2)) = \sqrt[7]{\frac{2 (x^{7} + 2 - 2)}{2}}$

Step 5.2.5

Simplify by subtracting numbers.

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Step 5.2.5.1

Subtract $2$ from $2$ .

$f^{- 1} (2 (x^{7} + 2)) = \sqrt[7]{\frac{2 (x^{7} + 0)}{2}}$

Step 5.2.5.2

Add $x^{7}$ and $0$ .

$f^{- 1} (2 (x^{7} + 2)) = \sqrt[7]{\frac{2 x^{7}}{2}}$

Step 5.2.6

Reduce the expression by cancelling the common factors.

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Step 5.2.6.1

Reduce the expression $\frac{2 x^{7}}{2}$ by cancelling the common factors.

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Step 5.2.6.1.1

Cancel the common factor.

$f^{- 1} (2 (x^{7} + 2)) = \sqrt[7]{\frac{2 x^{7}}{2}}$

Step 5.2.6.1.2

Rewrite the expression.

$f^{- 1} (2 (x^{7} + 2)) = \sqrt[7]{\frac{x^{7}}{1}}$

Step 5.2.6.2

Divide $x^{7}$ by $1$ .

$f^{- 1} (2 (x^{7} + 2)) = \sqrt[7]{x^{7}}$

Step 5.2.7

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

$f^{- 1} (2 (x^{7} + 2)) = x$

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 $f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 ({(\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}})}^{7} + 2)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the product rule to $\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}$ .

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{{\sqrt[7]{x - 4}}^{7}}{{\sqrt[7]{2}}^{7}} + 2)$

Step 5.3.3.2

Rewrite ${\sqrt[7]{x - 4}}^{7}$ as $x - 4$ .

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Step 5.3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x - 4}$ as ${(x - 4)}^{\frac{1}{7}}$ .

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{{({(x - 4)}^{\frac{1}{7}})}^{7}}{{\sqrt[7]{2}}^{7}} + 2)$

Step 5.3.3.2.2

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

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{{(x - 4)}^{\frac{1}{7} \cdot 7}}{{\sqrt[7]{2}}^{7}} + 2)$

Step 5.3.3.2.3

Combine $\frac{1}{7}$ and $7$ .

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{{(x - 4)}^{\frac{7}{7}}}{{\sqrt[7]{2}}^{7}} + 2)$

Step 5.3.3.2.4

Cancel the common factor of $7$ .

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Step 5.3.3.2.4.1

Cancel the common factor.

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{{(x - 4)}^{\frac{7}{7}}}{{\sqrt[7]{2}}^{7}} + 2)$

Step 5.3.3.2.4.2

Rewrite the expression.

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4}{{\sqrt[7]{2}}^{7}} + 2)$

Step 5.3.3.2.5

Simplify.

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4}{{\sqrt[7]{2}}^{7}} + 2)$

Step 5.3.3.3

Rewrite ${\sqrt[7]{2}}^{7}$ as $2$ .

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Step 5.3.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{2}$ as $2^{\frac{1}{7}}$ .

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4}{{(2^{\frac{1}{7}})}^{7}} + 2)$

Step 5.3.3.3.2

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

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4}{2^{\frac{1}{7} \cdot 7}} + 2)$

Step 5.3.3.3.3

Combine $\frac{1}{7}$ and $7$ .

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4}{2^{\frac{7}{7}}} + 2)$

Step 5.3.3.3.4

Cancel the common factor of $7$ .

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Step 5.3.3.3.4.1

Cancel the common factor.

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4}{2^{\frac{7}{7}}} + 2)$

Step 5.3.3.3.4.2

Rewrite the expression.

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4}{2} + 2)$

Step 5.3.3.3.5

Evaluate the exponent.

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4}{2} + 2)$

Step 5.3.4

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

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4}{2} + 2 \cdot \frac{2}{2})$

Step 5.3.5

Simplify terms.

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Step 5.3.5.1

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

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4}{2} + \frac{2 \cdot 2}{2})$

Step 5.3.5.2

Combine the numerators over the common denominator.

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4 + 2 \cdot 2}{2})$

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Multiply $2$ by $2$ .

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x - 4 + 4}{2})$

Step 5.3.6.2

Add $- 4$ and $4$ .

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x + 0}{2})$

Step 5.3.6.3

Add $x$ and $0$ .

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x}{2})$

Step 5.3.7

Cancel the common factor of $2$ .

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Step 5.3.7.1

Cancel the common factor.

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = 2 (\frac{x}{2})$

Step 5.3.7.2

Rewrite the expression.

$f (\frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}$ is the inverse of $f (x) = 2 (x^{7} + 2)$ .

$f^{- 1} (x) = \frac{\sqrt[7]{x - 4}}{\sqrt[7]{2}}$