Precalculus Examples

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

Step 1

Write $f (x) = 6 x^{5} - 1$ as an equation.

$y = 6 x^{5} - 1$

Step 2

Interchange the variables.

$x = 6 y^{5} - 1$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $6 y^{5} - 1 = x$ .

$6 y^{5} - 1 = x$

Step 3.2

Add $1$ to both sides of the equation.

$6 y^{5} = x + 1$

Step 3.3

Divide each term in $6 y^{5} = x + 1$ by $6$ and simplify.

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Step 3.3.1

Divide each term in $6 y^{5} = x + 1$ by $6$ .

$\frac{6 y^{5}}{6} = \frac{x}{6} + \frac{1}{6}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $6$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{6 y^{5}}{6} = \frac{x}{6} + \frac{1}{6}$

Step 3.3.2.1.2

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

$y^{5} = \frac{x}{6} + \frac{1}{6}$

Step 3.4

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

$y = \sqrt[5]{\frac{x}{6} + \frac{1}{6}}$

Step 3.5

Simplify $\sqrt[5]{\frac{x}{6} + \frac{1}{6}}$ .

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Step 3.5.1

Combine the numerators over the common denominator.

$y = \sqrt[5]{\frac{x + 1}{6}}$

Step 3.5.2

Rewrite $\sqrt[5]{\frac{x + 1}{6}}$ as $\frac{\sqrt[5]{x + 1}}{\sqrt[5]{6}}$ .

$y = \frac{\sqrt[5]{x + 1}}{\sqrt[5]{6}}$

Step 3.5.3

Multiply $\frac{\sqrt[5]{x + 1}}{\sqrt[5]{6}}$ by $\frac{{\sqrt[5]{6}}^{4}}{{\sqrt[5]{6}}^{4}}$ .

$y = \frac{\sqrt[5]{x + 1}}{\sqrt[5]{6}} \cdot \frac{{\sqrt[5]{6}}^{4}}{{\sqrt[5]{6}}^{4}}$

Step 3.5.4

Combine and simplify the denominator.

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Step 3.5.4.1

Multiply $\frac{\sqrt[5]{x + 1}}{\sqrt[5]{6}}$ by $\frac{{\sqrt[5]{6}}^{4}}{{\sqrt[5]{6}}^{4}}$ .

$y = \frac{\sqrt[5]{x + 1} {\sqrt[5]{6}}^{4}}{\sqrt[5]{6} {\sqrt[5]{6}}^{4}}$

Step 3.5.4.2

Raise $\sqrt[5]{6}$ to the power of $1$ .

$y = \frac{\sqrt[5]{x + 1} {\sqrt[5]{6}}^{4}}{{\sqrt[5]{6}}^{1} {\sqrt[5]{6}}^{4}}$

Step 3.5.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{\sqrt[5]{x + 1} {\sqrt[5]{6}}^{4}}{{\sqrt[5]{6}}^{1 + 4}}$

Step 3.5.4.4

Add $1$ and $4$ .

$y = \frac{\sqrt[5]{x + 1} {\sqrt[5]{6}}^{4}}{{\sqrt[5]{6}}^{5}}$

Step 3.5.4.5

Rewrite ${\sqrt[5]{6}}^{5}$ as $6$ .

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Step 3.5.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{6}$ as $6^{\frac{1}{5}}$ .

$y = \frac{\sqrt[5]{x + 1} {\sqrt[5]{6}}^{4}}{{(6^{\frac{1}{5}})}^{5}}$

Step 3.5.4.5.2

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

$y = \frac{\sqrt[5]{x + 1} {\sqrt[5]{6}}^{4}}{6^{\frac{1}{5} \cdot 5}}$

Step 3.5.4.5.3

Combine $\frac{1}{5}$ and $5$ .

$y = \frac{\sqrt[5]{x + 1} {\sqrt[5]{6}}^{4}}{6^{\frac{5}{5}}}$

Step 3.5.4.5.4

Cancel the common factor of $5$ .

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Step 3.5.4.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[5]{x + 1} {\sqrt[5]{6}}^{4}}{6^{\frac{5}{5}}}$

Step 3.5.4.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[5]{x + 1} {\sqrt[5]{6}}^{4}}{6^{1}}$

Step 3.5.4.5.5

Evaluate the exponent.

$y = \frac{\sqrt[5]{x + 1} {\sqrt[5]{6}}^{4}}{6}$

Step 3.5.5

Simplify the numerator.

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Step 3.5.5.1

Rewrite ${\sqrt[5]{6}}^{4}$ as $\sqrt[5]{6^{4}}$ .

$y = \frac{\sqrt[5]{x + 1} \sqrt[5]{6^{4}}}{6}$

Step 3.5.5.2

Raise $6$ to the power of $4$ .

$y = \frac{\sqrt[5]{x + 1} \sqrt[5]{1296}}{6}$

Step 3.5.6

Simplify with factoring out.

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Step 3.5.6.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[5]{(x + 1) \cdot 1296}}{6}$

Step 3.5.6.2

Reorder factors in $\frac{\sqrt[5]{(x + 1) \cdot 1296}}{6}$ .

$y = \frac{\sqrt[5]{1296 (x + 1)}}{6}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[5]{1296 (x + 1)}}{6}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[5]{1296 (x + 1)}}{6}$ is the inverse of $f (x) = 6 x^{5} - 1$ .

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

$f^{- 1} (6 x^{5} - 1) = \frac{\sqrt[5]{1296 ((6 x^{5} - 1) + 1)}}{6}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Add $- 1$ and $1$ .

$f^{- 1} (6 x^{5} - 1) = \frac{\sqrt[5]{1296 (6 x^{5} + 0)}}{6}$

Step 5.2.3.2

Add $6 x^{5}$ and $0$ .

$f^{- 1} (6 x^{5} - 1) = \frac{\sqrt[5]{1296 \cdot (6 x^{5})}}{6}$

Step 5.2.3.3

Multiply $1296$ by $6$ .

$f^{- 1} (6 x^{5} - 1) = \frac{\sqrt[5]{7776 x^{5}}}{6}$

Step 5.2.3.4

Rewrite $7776 x^{5}$ as ${(6 x)}^{5}$ .

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

Step 5.2.3.5

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

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

Step 5.2.4

Cancel the common factor of $6$ .

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Step 5.2.4.1

Cancel the common factor.

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

Step 5.2.4.2

Divide $x$ by $1$ .

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

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 $f (\frac{\sqrt[5]{1296 (x + 1)}}{6})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 {(\frac{\sqrt[5]{1296 (x + 1)}}{6})}^{5} - 1$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the product rule to $\frac{\sqrt[5]{1296 (x + 1)}}{6}$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{{\sqrt[5]{1296 (x + 1)}}^{5}}{6^{5}}) - 1$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

Rewrite ${\sqrt[5]{1296 (x + 1)}}^{5}$ as $1296 (x + 1)$ .

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Step 5.3.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{1296 (x + 1)}$ as ${(1296 (x + 1))}^{\frac{1}{5}}$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{{({(1296 (x + 1))}^{\frac{1}{5}})}^{5}}{6^{5}}) - 1$

Step 5.3.3.2.1.2

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

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{{(1296 (x + 1))}^{\frac{1}{5} \cdot 5}}{6^{5}}) - 1$

Step 5.3.3.2.1.3

Combine $\frac{1}{5}$ and $5$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{{(1296 (x + 1))}^{\frac{5}{5}}}{6^{5}}) - 1$

Step 5.3.3.2.1.4

Cancel the common factor of $5$ .

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Step 5.3.3.2.1.4.1

Cancel the common factor.

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{{(1296 (x + 1))}^{\frac{5}{5}}}{6^{5}}) - 1$

Step 5.3.3.2.1.4.2

Rewrite the expression.

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{1296 (x + 1)}{6^{5}}) - 1$

Step 5.3.3.2.1.5

Simplify.

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{1296 (x + 1)}{6^{5}}) - 1$

Step 5.3.3.2.2

Apply the distributive property.

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{1296 x + 1296 \cdot 1}{6^{5}}) - 1$

Step 5.3.3.2.3

Multiply $1296$ by $1$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{1296 x + 1296}{6^{5}}) - 1$

Step 5.3.3.2.4

Factor $1296$ out of $1296 x + 1296$ .

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Step 5.3.3.2.4.1

Factor $1296$ out of $1296 x$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{1296 (x) + 1296}{6^{5}}) - 1$

Step 5.3.3.2.4.2

Factor $1296$ out of $1296$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{1296 (x) + 1296 (1)}{6^{5}}) - 1$

Step 5.3.3.2.4.3

Factor $1296$ out of $1296 (x) + 1296 (1)$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{1296 (x + 1)}{6^{5}}) - 1$

Step 5.3.3.3

Raise $6$ to the power of $5$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{1296 (x + 1)}{7776}) - 1$

Step 5.3.3.4

Cancel the common factor of $6$ .

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Step 5.3.3.4.1

Factor $6$ out of $7776$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{1296 (x + 1)}{6 (1296)}) - 1$

Step 5.3.3.4.2

Cancel the common factor.

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = 6 (\frac{1296 (x + 1)}{6 \cdot 1296}) - 1$

Step 5.3.3.4.3

Rewrite the expression.

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = \frac{1296 (x + 1)}{1296} - 1$

Step 5.3.3.5

Cancel the common factor of $1296$ .

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Step 5.3.3.5.1

Cancel the common factor.

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = \frac{1296 (x + 1)}{1296} - 1$

Step 5.3.3.5.2

Divide $x + 1$ by $1$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = x + 1 - 1$

Step 5.3.4

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

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Step 5.3.4.1

Subtract $1$ from $1$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{\sqrt[5]{1296 (x + 1)}}{6}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[5]{1296 (x + 1)}}{6}$ is the inverse of $f (x) = 6 x^{5} - 1$ .

$f^{- 1} (x) = \frac{\sqrt[5]{1296 (x + 1)}}{6}$