Precalculus Examples

$f (x) = 8 x^{5}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = 8 x^{5}$ as an equation.

$y = 8 x^{5}$

Step 2

Interchange the variables.

$x = 8 y^{5}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $8 y^{5} = x$ .

$8 y^{5} = x$

Step 3.2

Divide each term in $8 y^{5} = x$ by $8$ and simplify.

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Step 3.2.1

Divide each term in $8 y^{5} = x$ by $8$ .

$\frac{8 y^{5}}{8} = \frac{x}{8}$

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.

$\frac{8 y^{5}}{8} = \frac{x}{8}$

Step 3.2.2.1.2

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

$y^{5} = \frac{x}{8}$

Step 3.3

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

$y = \sqrt[5]{\frac{x}{8}}$

Step 3.4

Simplify $\sqrt[5]{\frac{x}{8}}$ .

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Step 3.4.1

Rewrite $\sqrt[5]{\frac{x}{8}}$ as $\frac{\sqrt[5]{x}}{\sqrt[5]{8}}$ .

$y = \frac{\sqrt[5]{x}}{\sqrt[5]{8}}$

Step 3.4.2

Multiply $\frac{\sqrt[5]{x}}{\sqrt[5]{8}}$ by $\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$ .

$y = \frac{\sqrt[5]{x}}{\sqrt[5]{8}} \cdot \frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$

Step 3.4.3

Combine and simplify the denominator.

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Step 3.4.3.1

Multiply $\frac{\sqrt[5]{x}}{\sqrt[5]{8}}$ by $\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$ .

$y = \frac{\sqrt[5]{x} {\sqrt[5]{8}}^{4}}{\sqrt[5]{8} {\sqrt[5]{8}}^{4}}$

Step 3.4.3.2

Raise $\sqrt[5]{8}$ to the power of $1$ .

$y = \frac{\sqrt[5]{x} {\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{1} {\sqrt[5]{8}}^{4}}$

Step 3.4.3.3

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

$y = \frac{\sqrt[5]{x} {\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{1 + 4}}$

Step 3.4.3.4

Add $1$ and $4$ .

$y = \frac{\sqrt[5]{x} {\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{5}}$

Step 3.4.3.5

Rewrite ${\sqrt[5]{8}}^{5}$ as $8$ .

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Step 3.4.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{8}$ as $8^{\frac{1}{5}}$ .

$y = \frac{\sqrt[5]{x} {\sqrt[5]{8}}^{4}}{{(8^{\frac{1}{5}})}^{5}}$

Step 3.4.3.5.2

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

$y = \frac{\sqrt[5]{x} {\sqrt[5]{8}}^{4}}{8^{\frac{1}{5} \cdot 5}}$

Step 3.4.3.5.3

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

$y = \frac{\sqrt[5]{x} {\sqrt[5]{8}}^{4}}{8^{\frac{5}{5}}}$

Step 3.4.3.5.4

Cancel the common factor of $5$ .

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Step 3.4.3.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[5]{x} {\sqrt[5]{8}}^{4}}{8^{\frac{5}{5}}}$

Step 3.4.3.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[5]{x} {\sqrt[5]{8}}^{4}}{8^{1}}$

Step 3.4.3.5.5

Evaluate the exponent.

$y = \frac{\sqrt[5]{x} {\sqrt[5]{8}}^{4}}{8}$

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Rewrite ${\sqrt[5]{8}}^{4}$ as $\sqrt[5]{8^{4}}$ .

$y = \frac{\sqrt[5]{x} \sqrt[5]{8^{4}}}{8}$

Step 3.4.4.2

Raise $8$ to the power of $4$ .

$y = \frac{\sqrt[5]{x} \sqrt[5]{4096}}{8}$

Step 3.4.4.3

Rewrite $4096$ as $4^{5} \cdot 4$ .

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Step 3.4.4.3.1

Factor $1024$ out of $4096$ .

$y = \frac{\sqrt[5]{x} \sqrt[5]{1024 (4)}}{8}$

Step 3.4.4.3.2

Rewrite $1024$ as $4^{5}$ .

$y = \frac{\sqrt[5]{x} \sqrt[5]{4^{5} \cdot 4}}{8}$

Step 3.4.4.4

Pull terms out from under the radical.

$y = \frac{\sqrt[5]{x} \cdot 4 \sqrt[5]{4}}{8}$

Step 3.4.4.5

Combine using the product rule for radicals.

$y = \frac{4 \sqrt[5]{x \cdot 4}}{8}$

Step 3.4.5

Reduce the expression by cancelling the common factors.

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Step 3.4.5.1

Cancel the common factor of $4$ and $8$ .

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Step 3.4.5.1.1

Factor $4$ out of $4 \sqrt[5]{x \cdot 4}$ .

$y = \frac{4 (\sqrt[5]{x \cdot 4})}{8}$

Step 3.4.5.1.2

Cancel the common factors.

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Step 3.4.5.1.2.1

Factor $4$ out of $8$ .

$y = \frac{4 \sqrt[5]{x \cdot 4}}{4 \cdot 2}$

Step 3.4.5.1.2.2

Cancel the common factor.

$y = \frac{4 \sqrt[5]{x \cdot 4}}{4 \cdot 2}$

Step 3.4.5.1.2.3

Rewrite the expression.

$y = \frac{\sqrt[5]{x \cdot 4}}{2}$

Step 3.4.5.2

Reorder factors in $\frac{\sqrt[5]{x \cdot 4}}{2}$ .

$y = \frac{\sqrt[5]{4 x}}{2}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[5]{4 x}}{2}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[5]{4 x}}{2}$ is the inverse of $f (x) = 8 x^{5}$ .

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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 $f^{- 1} (8 x^{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (8 x^{5}) = \frac{\sqrt[5]{4 (8 x^{5})}}{2}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Multiply $4$ by $8$ .

$f^{- 1} (8 x^{5}) = \frac{\sqrt[5]{32 x^{5}}}{2}$

Step 5.2.3.2

Rewrite $32 x^{5}$ as ${(2 x)}^{5}$ .

$f^{- 1} (8 x^{5}) = \frac{\sqrt[5]{{(2 x)}^{5}}}{2}$

Step 5.2.3.3

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

$f^{- 1} (8 x^{5}) = \frac{2 x}{2}$

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (8 x^{5}) = \frac{2 x}{2}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (8 x^{5}) = x$

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 $f (\frac{\sqrt[5]{4 x}}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 {(\frac{\sqrt[5]{4 x}}{2})}^{5}$

Step 5.3.3

Apply the product rule to $\frac{\sqrt[5]{4 x}}{2}$ .

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 (\frac{{\sqrt[5]{4 x}}^{5}}{2^{5}})$

Step 5.3.4

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

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Step 5.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4 x}$ as ${(4 x)}^{\frac{1}{5}}$ .

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 (\frac{{({(4 x)}^{\frac{1}{5}})}^{5}}{2^{5}})$

Step 5.3.4.2

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

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 (\frac{{(4 x)}^{\frac{1}{5} \cdot 5}}{2^{5}})$

Step 5.3.4.3

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

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 (\frac{{(4 x)}^{\frac{5}{5}}}{2^{5}})$

Step 5.3.4.4

Cancel the common factor of $5$ .

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Step 5.3.4.4.1

Cancel the common factor.

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 (\frac{{(4 x)}^{\frac{5}{5}}}{2^{5}})$

Step 5.3.4.4.2

Rewrite the expression.

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 (\frac{4 x}{2^{5}})$

Step 5.3.4.5

Simplify.

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 (\frac{4 x}{2^{5}})$

Step 5.3.5

Raise $2$ to the power of $5$ .

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 (\frac{4 x}{32})$

Step 5.3.6

Cancel the common factor of $8$ .

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Step 5.3.6.1

Factor $8$ out of $32$ .

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 (\frac{4 x}{8 (4)})$

Step 5.3.6.2

Cancel the common factor.

$f (\frac{\sqrt[5]{4 x}}{2}) = 8 (\frac{4 x}{8 \cdot 4})$

Step 5.3.6.3

Rewrite the expression.

$f (\frac{\sqrt[5]{4 x}}{2}) = \frac{4 x}{4}$

Step 5.3.7

Cancel the common factor of $4$ .

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Step 5.3.7.1

Cancel the common factor.

$f (\frac{\sqrt[5]{4 x}}{2}) = \frac{4 x}{4}$

Step 5.3.7.2

Divide $x$ by $1$ .

$f (\frac{\sqrt[5]{4 x}}{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[5]{4 x}}{2}$ is the inverse of $f (x) = 8 x^{5}$ .

$f^{- 1} (x) = \frac{\sqrt[5]{4 x}}{2}$