Precalculus Examples

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

Step 1

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

$y = 5 {(x - 5)}^{7}$

Step 2

Interchange the variables.

$x = 5 {(y - 5)}^{7}$

Step 3

Solve for $y$ .

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Step 3.1

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

$5 {(y - 5)}^{7} = x$

Step 3.2

Divide each term in $5 {(y - 5)}^{7} = x$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 {(y - 5)}^{7} = x$ by $5$ .

$\frac{5 {(y - 5)}^{7}}{5} = \frac{x}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 {(y - 5)}^{7}}{5} = \frac{x}{5}$

Step 3.2.2.1.2

Divide ${(y - 5)}^{7}$ by $1$ .

${(y - 5)}^{7} = \frac{x}{5}$

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $5$ to both sides of the equation.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Combine $\sqrt[7]{5 {(x - 5)}^{7}}$ and $\sqrt[7]{5}$ into a single radical.

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

Step 5.2.3.2

Reduce the expression by cancelling the common factors.

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Step 5.2.3.2.1

Reduce the expression $\frac{5 {(x - 5)}^{7}}{5}$ by cancelling the common factors.

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Step 5.2.3.2.1.1

Cancel the common factor.

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

Step 5.2.3.2.1.2

Rewrite the expression.

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

Step 5.2.3.2.2

Divide ${(x - 5)}^{7}$ by $1$ .

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

Step 5.2.3.3

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

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

Step 5.2.4

Combine the opposite terms in $x - 5 + 5$ .

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Step 5.2.4.1

Add $- 5$ and $5$ .

$f^{- 1} (5 {(x - 5)}^{7}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

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

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

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

Step 5.3.3

Combine the opposite terms in $(\frac{\sqrt[7]{x}}{\sqrt[7]{5}} + 5) - 5$ .

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Step 5.3.3.1

Subtract $5$ from $5$ .

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

Step 5.3.3.2

Add $\frac{\sqrt[7]{x}}{\sqrt[7]{5}}$ and $0$ .

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

Step 5.3.4

Apply the product rule to $\frac{\sqrt[7]{x}}{\sqrt[7]{5}}$ .

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

Step 5.3.5

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

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Step 5.3.5.1

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

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

Step 5.3.5.2

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

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

Step 5.3.5.3

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

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

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.

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

Step 5.3.5.4.2

Rewrite the expression.

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

Step 5.3.5.5

Simplify.

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

Step 5.3.6

Rewrite ${\sqrt[7]{5}}^{7}$ as $5$ .

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Step 5.3.6.1

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

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

Step 5.3.6.2

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

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

Step 5.3.6.3

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

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

Step 5.3.6.4

Cancel the common factor of $7$ .

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Step 5.3.6.4.1

Cancel the common factor.

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

Step 5.3.6.4.2

Rewrite the expression.

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

Step 5.3.6.5

Evaluate the exponent.

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

Step 5.3.7

Cancel the common factor of $5$ .

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Step 5.3.7.1

Cancel the common factor.

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

Step 5.3.7.2

Rewrite the expression.

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

Step 5.4

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

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