Precalculus Examples

$f (x) = 8 {(x + 9)}^{\frac{1}{7}}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = 8 {(x + 9)}^{\frac{1}{7}}$ as an equation.

$y = 8 {(x + 9)}^{\frac{1}{7}}$

Step 2

Interchange the variables.

$x = 8 {(y + 9)}^{\frac{1}{7}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $8 {(y + 9)}^{\frac{1}{7}} = x$ .

$8 {(y + 9)}^{\frac{1}{7}} = x$

Step 3.2

Divide each term in $8 {(y + 9)}^{\frac{1}{7}} = x$ by $8$ and simplify.

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Step 3.2.1

Divide each term in $8 {(y + 9)}^{\frac{1}{7}} = x$ by $8$ .

$\frac{8 {(y + 9)}^{\frac{1}{7}}}{8} = \frac{x}{8}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor.

$\frac{8 {(y + 9)}^{\frac{1}{7}}}{8} = \frac{x}{8}$

Step 3.2.2.2

Divide ${(y + 9)}^{\frac{1}{7}}$ by $1$ .

${(y + 9)}^{\frac{1}{7}} = \frac{x}{8}$

Step 3.3

Raise each side of the equation to the power of $7$ to eliminate the fractional exponent on the left side.

${({(y + 9)}^{\frac{1}{7}})}^{7} = {(\frac{x}{8})}^{7}$

Step 3.4

Simplify the exponent.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify ${({(y + 9)}^{\frac{1}{7}})}^{7}$ .

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Step 3.4.1.1.1

Multiply the exponents in ${({(y + 9)}^{\frac{1}{7}})}^{7}$ .

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Step 3.4.1.1.1.1

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

${(y + 9)}^{\frac{1}{7} \cdot 7} = {(\frac{x}{8})}^{7}$

Step 3.4.1.1.1.2

Cancel the common factor of $7$ .

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Step 3.4.1.1.1.2.1

Cancel the common factor.

${(y + 9)}^{\frac{1}{7} \cdot 7} = {(\frac{x}{8})}^{7}$

Step 3.4.1.1.1.2.2

Rewrite the expression.

${(y + 9)}^{1} = {(\frac{x}{8})}^{7}$

Step 3.4.1.1.2

Simplify.

$y + 9 = {(\frac{x}{8})}^{7}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify ${(\frac{x}{8})}^{7}$ .

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Step 3.4.2.1.1

Apply the product rule to $\frac{x}{8}$ .

$y + 9 = \frac{x^{7}}{8^{7}}$

Step 3.4.2.1.2

Raise $8$ to the power of $7$ .

$y + 9 = \frac{x^{7}}{2097152}$

Step 3.5

Subtract $9$ from both sides of the equation.

$y = \frac{x^{7}}{2097152} - 9$

Step 4

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

$f^{- 1} (x) = \frac{x^{7}}{2097152} - 9$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{7}}{2097152} - 9$ is the inverse of $f (x) = 8 {(x + 9)}^{\frac{1}{7}}$ .

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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 $f^{- 1} (8 {(x + 9)}^{\frac{1}{7}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = \frac{{(8 {(x + 9)}^{\frac{1}{7}})}^{7}}{2097152} - 9$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Apply the product rule to $8 {(x + 9)}^{\frac{1}{7}}$ .

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = \frac{8^{7} {({(x + 9)}^{\frac{1}{7}})}^{7}}{2097152} - 9$

Step 5.2.3.1.2

Raise $8$ to the power of $7$ .

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = \frac{2097152 {({(x + 9)}^{\frac{1}{7}})}^{7}}{2097152} - 9$

Step 5.2.3.1.3

Multiply the exponents in ${({(x + 9)}^{\frac{1}{7}})}^{7}$ .

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Step 5.2.3.1.3.1

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

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = \frac{2097152 {(x + 9)}^{\frac{1}{7} \cdot 7}}{2097152} - 9$

Step 5.2.3.1.3.2

Cancel the common factor of $7$ .

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Step 5.2.3.1.3.2.1

Cancel the common factor.

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = \frac{2097152 {(x + 9)}^{\frac{1}{7} \cdot 7}}{2097152} - 9$

Step 5.2.3.1.3.2.2

Rewrite the expression.

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = \frac{2097152 (x + 9)}{2097152} - 9$

Step 5.2.3.1.4

Simplify.

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = \frac{2097152 (x + 9)}{2097152} - 9$

Step 5.2.3.2

Cancel the common factor of $2097152$ .

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Step 5.2.3.2.1

Cancel the common factor.

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = \frac{2097152 (x + 9)}{2097152} - 9$

Step 5.2.3.2.2

Divide $x + 9$ by $1$ .

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = x + 9 - 9$

Step 5.2.4

Combine the opposite terms in $x + 9 - 9$ .

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Step 5.2.4.1

Subtract $9$ from $9$ .

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (8 {(x + 9)}^{\frac{1}{7}}) = x$

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 $f (\frac{x^{7}}{2097152} - 9)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{7}}{2097152} - 9) = 8 {((\frac{x^{7}}{2097152} - 9) + 9)}^{\frac{1}{7}}$

Step 5.3.3

Simplify by adding terms.

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Step 5.3.3.1

Combine the opposite terms in $(\frac{x^{7}}{2097152} - 9) + 9$ .

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Step 5.3.3.1.1

Add $- 9$ and $9$ .

$f (\frac{x^{7}}{2097152} - 9) = 8 {(\frac{x^{7}}{2097152} + 0)}^{\frac{1}{7}}$

Step 5.3.3.1.2

Add $\frac{x^{7}}{2097152}$ and $0$ .

$f (\frac{x^{7}}{2097152} - 9) = 8 {(\frac{x^{7}}{2097152})}^{\frac{1}{7}}$

Step 5.3.3.2

Apply the product rule to $\frac{x^{7}}{2097152}$ .

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{{(x^{7})}^{\frac{1}{7}}}{2097152^{\frac{1}{7}}})$

Step 5.3.4

Simplify the numerator.

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Step 5.3.4.1

Multiply the exponents in ${(x^{7})}^{\frac{1}{7}}$ .

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Step 5.3.4.1.1

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

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{x^{7 (\frac{1}{7})}}{2097152^{\frac{1}{7}}})$

Step 5.3.4.1.2

Cancel the common factor of $7$ .

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Step 5.3.4.1.2.1

Cancel the common factor.

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{x^{7 (\frac{1}{7})}}{2097152^{\frac{1}{7}}})$

Step 5.3.4.1.2.2

Rewrite the expression.

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{x}{2097152^{\frac{1}{7}}})$

Step 5.3.4.2

Simplify.

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{x}{2097152^{\frac{1}{7}}})$

Step 5.3.5

Simplify the denominator.

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Step 5.3.5.1

Rewrite $2097152$ as $8^{7}$ .

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{x}{{(8^{7})}^{\frac{1}{7}}})$

Step 5.3.5.2

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

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{x}{8^{7 (\frac{1}{7})}})$

Step 5.3.5.3

Cancel the common factor of $7$ .

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Step 5.3.5.3.1

Cancel the common factor.

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{x}{8^{7 (\frac{1}{7})}})$

Step 5.3.5.3.2

Rewrite the expression.

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{x}{8})$

Step 5.3.5.4

Evaluate the exponent.

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{x}{8})$

Step 5.3.6

Cancel the common factor of $8$ .

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Step 5.3.6.1

Cancel the common factor.

$f (\frac{x^{7}}{2097152} - 9) = 8 (\frac{x}{8})$

Step 5.3.6.2

Rewrite the expression.

$f (\frac{x^{7}}{2097152} - 9) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{7}}{2097152} - 9$ is the inverse of $f (x) = 8 {(x + 9)}^{\frac{1}{7}}$ .

$f^{- 1} (x) = \frac{x^{7}}{2097152} - 9$