Precalculus Examples

$f (x) = \frac{x^{5} - 6}{9}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \frac{x^{5} - 6}{9}$ as an equation.

$y = \frac{x^{5} - 6}{9}$

Step 2

Interchange the variables.

$x = \frac{y^{5} - 6}{9}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y^{5} - 6}{9} = x$ .

$\frac{y^{5} - 6}{9} = x$

Step 3.2

Multiply both sides of the equation by $9$ .

$9 \frac{y^{5} - 6}{9} = 9 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $9$ .

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Step 3.3.1.1

Cancel the common factor.

$9 \frac{y^{5} - 6}{9} = 9 x$

Step 3.3.1.2

Rewrite the expression.

$y^{5} - 6 = 9 x$

Step 3.4

Add $6$ to both sides of the equation.

$y^{5} = 9 x + 6$

Step 3.5

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

$y = \sqrt[5]{9 x + 6}$

Step 3.6

Factor $3$ out of $9 x + 6$ .

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Step 3.6.1

Factor $3$ out of $9 x$ .

$y = \sqrt[5]{3 (3 x) + 6}$

Step 3.6.2

Factor $3$ out of $6$ .

$y = \sqrt[5]{3 (3 x) + 3 (2)}$

Step 3.6.3

Factor $3$ out of $3 (3 x) + 3 (2)$ .

$y = \sqrt[5]{3 (3 x + 2)}$

Step 4

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

$f^{- 1} (x) = \sqrt[5]{3 (3 x + 2)}$

Step 5

Verify if $f^{- 1} (x) = \sqrt[5]{3 (3 x + 2)}$ is the inverse of $f (x) = \frac{x^{5} - 6}{9}$ .

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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 $f^{- 1} (\frac{x^{5} - 6}{9})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (3 (\frac{x^{5} - 6}{9}) + 2)}$

Step 5.2.3

Cancel the common factor of $3$ .

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Step 5.2.3.1

Factor $3$ out of $9$ .

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (3 (\frac{x^{5} - 6}{3 (3)}) + 2)}$

Step 5.2.3.2

Cancel the common factor.

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (3 (\frac{x^{5} - 6}{3 \cdot 3}) + 2)}$

Step 5.2.3.3

Rewrite the expression.

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (\frac{x^{5} - 6}{3} + 2)}$

Step 5.2.4

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (\frac{x^{5} - 6}{3} + 2 \cdot \frac{3}{3})}$

Step 5.2.5

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

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (\frac{x^{5} - 6}{3} + \frac{2 \cdot 3}{3})}$

Step 5.2.6

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (\frac{x^{5} - 6 + 2 \cdot 3}{3})}$

Step 5.2.7

Reorder terms.

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (\frac{x^{5} + 2 \cdot 3 - 6}{3})}$

Step 5.2.8

Rewrite $\frac{x^{5} + 2 \cdot 3 - 6}{3}$ in a factored form.

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Step 5.2.8.1

Multiply $2$ by $3$ .

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (\frac{x^{5} + 6 - 6}{3})}$

Step 5.2.8.2

Subtract $6$ from $6$ .

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (\frac{x^{5} + 0}{3})}$

Step 5.2.8.3

Add $x^{5}$ and $0$ .

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{3 (\frac{x^{5}}{3})}$

Step 5.2.9

Combine $3$ and $\frac{x^{5}}{3}$ .

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{\frac{3 x^{5}}{3}}$

Step 5.2.10

Reduce the expression by cancelling the common factors.

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Step 5.2.10.1

Reduce the expression $\frac{3 x^{5}}{3}$ by cancelling the common factors.

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Step 5.2.10.1.1

Cancel the common factor.

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{\frac{3 x^{5}}{3}}$

Step 5.2.10.1.2

Rewrite the expression.

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{\frac{x^{5}}{1}}$

Step 5.2.10.2

Divide $x^{5}$ by $1$ .

$f^{- 1} (\frac{x^{5} - 6}{9}) = \sqrt[5]{x^{5}}$

Step 5.2.11

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

$f^{- 1} (\frac{x^{5} - 6}{9}) = x$

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 $f (\sqrt[5]{3 (3 x + 2)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{{(\sqrt[5]{3 (3 x + 2)})}^{5} - 6}{9}$

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 $\sqrt[5]{3 (3 x + 2)}$ as ${(3 (3 x + 2))}^{\frac{1}{5}}$ .

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{{({(3 (3 x + 2))}^{\frac{1}{5}})}^{5} - 6}{9}$

Step 5.3.3.2

Multiply the exponents in ${({(3 (3 x + 2))}^{\frac{1}{5}})}^{5}$ .

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Step 5.3.3.2.1

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

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{{(3 (3 x + 2))}^{\frac{1}{5} \cdot 5} - 6}{9}$

Step 5.3.3.2.2

Cancel the common factor of $5$ .

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Step 5.3.3.2.2.1

Cancel the common factor.

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{{(3 (3 x + 2))}^{\frac{1}{5} \cdot 5} - 6}{9}$

Step 5.3.3.2.2.2

Rewrite the expression.

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{(3 (3 x + 2)) - 6}{9}$

Step 5.3.3.3

Apply the distributive property.

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{(3 (3 x) + 3 \cdot 2) - 6}{9}$

Step 5.3.3.4

Multiply $3$ by $3$ .

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{(9 x + 3 \cdot 2) - 6}{9}$

Step 5.3.3.5

Multiply $3$ by $2$ .

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{(9 x + 6) - 6}{9}$

Step 5.3.3.6

Simplify.

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{9 x + 6 - 6}{9}$

Step 5.3.3.7

Subtract $6$ from $6$ .

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{9 x + 0}{9}$

Step 5.3.3.8

Add $9 x$ and $0$ .

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{9 x}{9}$

Step 5.3.4

Cancel the common factor of $9$ .

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Step 5.3.4.1

Cancel the common factor.

$f (\sqrt[5]{3 (3 x + 2)}) = \frac{9 x}{9}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (\sqrt[5]{3 (3 x + 2)}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[5]{3 (3 x + 2)}$ is the inverse of $f (x) = \frac{x^{5} - 6}{9}$ .

$f^{- 1} (x) = \sqrt[5]{3 (3 x + 2)}$