Precalculus Examples

$f (x) = 8 x^{7}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = 8 x^{7}$ as an equation.

$y = 8 x^{7}$

Step 2

Interchange the variables.

$x = 8 y^{7}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $8 y^{7} = x$ .

$8 y^{7} = x$

Step 3.2

Divide each term in $8 y^{7} = x$ by $8$ and simplify.

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Step 3.2.1

Divide each term in $8 y^{7} = x$ by $8$ .

$\frac{8 y^{7}}{8} = \frac{x}{8}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $8$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{8 y^{7}}{8} = \frac{x}{8}$

Step 3.2.2.1.2

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

$y^{7} = \frac{x}{8}$

Step 3.3

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

$y = \sqrt[7]{\frac{x}{8}}$

Step 3.4

Rewrite $\sqrt[7]{\frac{x}{8}}$ as $\frac{\sqrt[7]{x}}{\sqrt[7]{8}}$ .

$y = \frac{\sqrt[7]{x}}{\sqrt[7]{8}}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[7]{x}}{\sqrt[7]{8}}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[7]{x}}{\sqrt[7]{8}}$ is the inverse of $f (x) = 8 x^{7}$ .

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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 $f^{- 1} (8 x^{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (8 x^{7}) = \frac{\sqrt[7]{8 x^{7}}}{\sqrt[7]{8}}$

Step 5.2.3

Combine $\sqrt[7]{8 x^{7}}$ and $\sqrt[7]{8}$ into a single radical.

$f^{- 1} (8 x^{7}) = \sqrt[7]{\frac{8 x^{7}}{8}}$

Step 5.2.4

Reduce the expression by cancelling the common factors.

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Step 5.2.4.1

Reduce the expression $\frac{8 x^{7}}{8}$ by cancelling the common factors.

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Step 5.2.4.1.1

Cancel the common factor.

$f^{- 1} (8 x^{7}) = \sqrt[7]{\frac{8 x^{7}}{8}}$

Step 5.2.4.1.2

Rewrite the expression.

$f^{- 1} (8 x^{7}) = \sqrt[7]{\frac{x^{7}}{1}}$

Step 5.2.4.2

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

$f^{- 1} (8 x^{7}) = \sqrt[7]{x^{7}}$

Step 5.2.5

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

$f^{- 1} (8 x^{7}) = x$

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 $f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 {(\frac{\sqrt[7]{x}}{\sqrt[7]{8}})}^{7}$

Step 5.3.3

Apply the product rule to $\frac{\sqrt[7]{x}}{\sqrt[7]{8}}$ .

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{{\sqrt[7]{x}}^{7}}{{\sqrt[7]{8}}^{7}})$

Step 5.3.4

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

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Step 5.3.4.1

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

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{{(x^{\frac{1}{7}})}^{7}}{{\sqrt[7]{8}}^{7}})$

Step 5.3.4.2

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

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x^{\frac{1}{7} \cdot 7}}{{\sqrt[7]{8}}^{7}})$

Step 5.3.4.3

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

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x^{\frac{7}{7}}}{{\sqrt[7]{8}}^{7}})$

Step 5.3.4.4

Cancel the common factor of $7$ .

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Step 5.3.4.4.1

Cancel the common factor.

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x^{\frac{7}{7}}}{{\sqrt[7]{8}}^{7}})$

Step 5.3.4.4.2

Rewrite the expression.

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x}{{\sqrt[7]{8}}^{7}})$

Step 5.3.4.5

Simplify.

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x}{{\sqrt[7]{8}}^{7}})$

Step 5.3.5

Rewrite ${\sqrt[7]{8}}^{7}$ as $8$ .

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Step 5.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{8}$ as $8^{\frac{1}{7}}$ .

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x}{{(8^{\frac{1}{7}})}^{7}})$

Step 5.3.5.2

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

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x}{8^{\frac{1}{7} \cdot 7}})$

Step 5.3.5.3

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

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x}{8^{\frac{7}{7}}})$

Step 5.3.5.4

Cancel the common factor of $7$ .

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Step 5.3.5.4.1

Cancel the common factor.

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x}{8^{\frac{7}{7}}})$

Step 5.3.5.4.2

Rewrite the expression.

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x}{8})$

Step 5.3.5.5

Evaluate the exponent.

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x}{8})$

Step 5.3.6

Cancel the common factor of $8$ .

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Step 5.3.6.1

Cancel the common factor.

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = 8 (\frac{x}{8})$

Step 5.3.6.2

Rewrite the expression.

$f (\frac{\sqrt[7]{x}}{\sqrt[7]{8}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[7]{x}}{\sqrt[7]{8}}$ is the inverse of $f (x) = 8 x^{7}$ .

$f^{- 1} (x) = \frac{\sqrt[7]{x}}{\sqrt[7]{8}}$