Precalculus Examples

$f (x) = 8 x^{5} - 7$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = 8 x^{5} - 7$ as an equation.

$y = 8 x^{5} - 7$

Step 2

Interchange the variables.

$x = 8 y^{5} - 7$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $8 y^{5} - 7 = x$ .

$8 y^{5} - 7 = x$

Step 3.2

Add $7$ to both sides of the equation.

$8 y^{5} = x + 7$

Step 3.3

Divide each term in $8 y^{5} = x + 7$ by $8$ and simplify.

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Step 3.3.1

Divide each term in $8 y^{5} = x + 7$ by $8$ .

$\frac{8 y^{5}}{8} = \frac{x}{8} + \frac{7}{8}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $8$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{8 y^{5}}{8} = \frac{x}{8} + \frac{7}{8}$

Step 3.3.2.1.2

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

$y^{5} = \frac{x}{8} + \frac{7}{8}$

Step 3.4

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

$y = \sqrt[5]{\frac{x}{8} + \frac{7}{8}}$

Step 3.5

Simplify $\sqrt[5]{\frac{x}{8} + \frac{7}{8}}$ .

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Step 3.5.1

Combine the numerators over the common denominator.

$y = \sqrt[5]{\frac{x + 7}{8}}$

Step 3.5.2

Rewrite $\sqrt[5]{\frac{x + 7}{8}}$ as $\frac{\sqrt[5]{x + 7}}{\sqrt[5]{8}}$ .

$y = \frac{\sqrt[5]{x + 7}}{\sqrt[5]{8}}$

Step 3.5.3

Multiply $\frac{\sqrt[5]{x + 7}}{\sqrt[5]{8}}$ by $\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$ .

$y = \frac{\sqrt[5]{x + 7}}{\sqrt[5]{8}} \cdot \frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$

Step 3.5.4

Combine and simplify the denominator.

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Step 3.5.4.1

Multiply $\frac{\sqrt[5]{x + 7}}{\sqrt[5]{8}}$ by $\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$ .

$y = \frac{\sqrt[5]{x + 7} {\sqrt[5]{8}}^{4}}{\sqrt[5]{8} {\sqrt[5]{8}}^{4}}$

Step 3.5.4.2

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

$y = \frac{\sqrt[5]{x + 7} {\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{1} {\sqrt[5]{8}}^{4}}$

Step 3.5.4.3

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

$y = \frac{\sqrt[5]{x + 7} {\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{1 + 4}}$

Step 3.5.4.4

Add $1$ and $4$ .

$y = \frac{\sqrt[5]{x + 7} {\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{5}}$

Step 3.5.4.5

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

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Step 3.5.4.5.1

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

$y = \frac{\sqrt[5]{x + 7} {\sqrt[5]{8}}^{4}}{{(8^{\frac{1}{5}})}^{5}}$

Step 3.5.4.5.2

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

$y = \frac{\sqrt[5]{x + 7} {\sqrt[5]{8}}^{4}}{8^{\frac{1}{5} \cdot 5}}$

Step 3.5.4.5.3

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

$y = \frac{\sqrt[5]{x + 7} {\sqrt[5]{8}}^{4}}{8^{\frac{5}{5}}}$

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.

$y = \frac{\sqrt[5]{x + 7} {\sqrt[5]{8}}^{4}}{8^{\frac{5}{5}}}$

Step 3.5.4.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[5]{x + 7} {\sqrt[5]{8}}^{4}}{8^{1}}$

Step 3.5.4.5.5

Evaluate the exponent.

$y = \frac{\sqrt[5]{x + 7} {\sqrt[5]{8}}^{4}}{8}$

Step 3.5.5

Simplify the numerator.

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Step 3.5.5.1

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

$y = \frac{\sqrt[5]{x + 7} \sqrt[5]{8^{4}}}{8}$

Step 3.5.5.2

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

$y = \frac{\sqrt[5]{x + 7} \sqrt[5]{4096}}{8}$

Step 3.5.5.3

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

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Step 3.5.5.3.1

Factor $1024$ out of $4096$ .

$y = \frac{\sqrt[5]{x + 7} \sqrt[5]{1024 (4)}}{8}$

Step 3.5.5.3.2

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

$y = \frac{\sqrt[5]{x + 7} \sqrt[5]{4^{5} \cdot 4}}{8}$

Step 3.5.5.4

Pull terms out from under the radical.

$y = \frac{\sqrt[5]{x + 7} \cdot 4 \sqrt[5]{4}}{8}$

Step 3.5.5.5

Combine using the product rule for radicals.

$y = \frac{4 \sqrt[5]{(x + 7) \cdot 4}}{8}$

Step 3.5.6

Reduce the expression by cancelling the common factors.

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Step 3.5.6.1

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

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Step 3.5.6.1.1

Factor $4$ out of $4 \sqrt[5]{(x + 7) \cdot 4}$ .

$y = \frac{4 (\sqrt[5]{(x + 7) \cdot 4})}{8}$

Step 3.5.6.1.2

Cancel the common factors.

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Step 3.5.6.1.2.1

Factor $4$ out of $8$ .

$y = \frac{4 \sqrt[5]{(x + 7) \cdot 4}}{4 \cdot 2}$

Step 3.5.6.1.2.2

Cancel the common factor.

$y = \frac{4 \sqrt[5]{(x + 7) \cdot 4}}{4 \cdot 2}$

Step 3.5.6.1.2.3

Rewrite the expression.

$y = \frac{\sqrt[5]{(x + 7) \cdot 4}}{2}$

Step 3.5.6.2

Reorder factors in $\frac{\sqrt[5]{(x + 7) \cdot 4}}{2}$ .

$y = \frac{\sqrt[5]{4 (x + 7)}}{2}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt[5]{4 (x + 7)}}{2}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt[5]{4 (x + 7)}}{2}$ is the inverse of $f (x) = 8 x^{5} - 7$ .

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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 $f^{- 1} (8 x^{5} - 7)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (8 x^{5} - 7) = \frac{\sqrt[5]{4 ((8 x^{5} - 7) + 7)}}{2}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Add $- 7$ and $7$ .

$f^{- 1} (8 x^{5} - 7) = \frac{\sqrt[5]{4 (8 x^{5} + 0)}}{2}$

Step 5.2.3.2

Add $8 x^{5}$ and $0$ .

$f^{- 1} (8 x^{5} - 7) = \frac{\sqrt[5]{4 \cdot (8 x^{5})}}{2}$

Step 5.2.3.3

Multiply $4$ by $8$ .

$f^{- 1} (8 x^{5} - 7) = \frac{\sqrt[5]{32 x^{5}}}{2}$

Step 5.2.3.4

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

$f^{- 1} (8 x^{5} - 7) = \frac{\sqrt[5]{{(2 x)}^{5}}}{2}$

Step 5.2.3.5

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

$f^{- 1} (8 x^{5} - 7) = \frac{2 x}{2}$

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (8 x^{5} - 7) = \frac{2 x}{2}$

Step 5.2.4.2

Divide $x$ by $1$ .

$f^{- 1} (8 x^{5} - 7) = x$

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 $f (\frac{\sqrt[5]{4 (x + 7)}}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 {(\frac{\sqrt[5]{4 (x + 7)}}{2})}^{5} - 7$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the product rule to $\frac{\sqrt[5]{4 (x + 7)}}{2}$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{{\sqrt[5]{4 (x + 7)}}^{5}}{2^{5}}) - 7$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

Rewrite ${\sqrt[5]{4 (x + 7)}}^{5}$ as $4 (x + 7)$ .

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Step 5.3.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{4 (x + 7)}$ as ${(4 (x + 7))}^{\frac{1}{5}}$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{{({(4 (x + 7))}^{\frac{1}{5}})}^{5}}{2^{5}}) - 7$

Step 5.3.3.2.1.2

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

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{{(4 (x + 7))}^{\frac{1}{5} \cdot 5}}{2^{5}}) - 7$

Step 5.3.3.2.1.3

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

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{{(4 (x + 7))}^{\frac{5}{5}}}{2^{5}}) - 7$

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.

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{{(4 (x + 7))}^{\frac{5}{5}}}{2^{5}}) - 7$

Step 5.3.3.2.1.4.2

Rewrite the expression.

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{4 (x + 7)}{2^{5}}) - 7$

Step 5.3.3.2.1.5

Simplify.

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{4 (x + 7)}{2^{5}}) - 7$

Step 5.3.3.2.2

Apply the distributive property.

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{4 x + 4 \cdot 7}{2^{5}}) - 7$

Step 5.3.3.2.3

Multiply $4$ by $7$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{4 x + 28}{2^{5}}) - 7$

Step 5.3.3.2.4

Factor $4$ out of $4 x + 28$ .

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Step 5.3.3.2.4.1

Factor $4$ out of $4 x$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{4 (x) + 28}{2^{5}}) - 7$

Step 5.3.3.2.4.2

Factor $4$ out of $28$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{4 x + 4 \cdot 7}{2^{5}}) - 7$

Step 5.3.3.2.4.3

Factor $4$ out of $4 x + 4 \cdot 7$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{4 (x + 7)}{2^{5}}) - 7$

Step 5.3.3.3

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

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{4 (x + 7)}{32}) - 7$

Step 5.3.3.4

Cancel the common factor of $8$ .

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Step 5.3.3.4.1

Factor $8$ out of $32$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{4 (x + 7)}{8 (4)}) - 7$

Step 5.3.3.4.2

Cancel the common factor.

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = 8 (\frac{4 (x + 7)}{8 \cdot 4}) - 7$

Step 5.3.3.4.3

Rewrite the expression.

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = \frac{4 (x + 7)}{4} - 7$

Step 5.3.3.5

Cancel the common factor of $4$ .

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Step 5.3.3.5.1

Cancel the common factor.

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = \frac{4 (x + 7)}{4} - 7$

Step 5.3.3.5.2

Divide $x + 7$ by $1$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = x + 7 - 7$

Step 5.3.4

Combine the opposite terms in $x + 7 - 7$ .

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Step 5.3.4.1

Subtract $7$ from $7$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{\sqrt[5]{4 (x + 7)}}{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\sqrt[5]{4 (x + 7)}}{2}$ is the inverse of $f (x) = 8 x^{5} - 7$ .

$f^{- 1} (x) = \frac{\sqrt[5]{4 (x + 7)}}{2}$