Algebra Examples

$f (x) = {(\frac{\sqrt[7]{x}}{10})}^{5} + 1$

Step 1

Write $f (x) = {(\frac{\sqrt[7]{x}}{10})}^{5} + 1$ as an equation.

$y = {(\frac{\sqrt[7]{x}}{10})}^{5} + 1$

Step 2

Interchange the variables.

$x = {(\frac{\sqrt[7]{y}}{10})}^{5} + 1$

Step 3

Solve for $y$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the product rule to $\frac{\sqrt[7]{y}}{10}$ .

$x = \frac{{\sqrt[7]{y}}^{5}}{10^{5}} + 1$

Step 3.1.2

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

$x = \frac{\sqrt[7]{y^{5}}}{10^{5}} + 1$

Step 3.1.3

Raise $10$ to the power of $5$ .

$x = \frac{\sqrt[7]{y^{5}}}{100000} + 1$

Step 3.2

Rewrite the equation as $\frac{\sqrt[7]{y^{5}}}{100000} + 1 = x$ .

$\frac{\sqrt[7]{y^{5}}}{100000} + 1 = x$

Step 3.3

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

$\frac{y^{\frac{5}{7}}}{100000} + 1 = x$

Step 3.4

Move all terms not containing $y$ to the right side of the equation.

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Step 3.4.1

Subtract $1$ from both sides of the equation.

$\frac{y^{\frac{5}{7}}}{100000} - x = - 1$

Step 3.4.2

Add $x$ to both sides of the equation.

$\frac{y^{\frac{5}{7}}}{100000} = - 1 + x$

Step 3.5

Multiply both sides of the equation by $100000$ .

$100000 \frac{y^{\frac{5}{7}}}{100000} = 100000 (- 1 + x)$

Step 3.6

Simplify both sides of the equation.

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Step 3.6.1

Simplify the left side.

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Step 3.6.1.1

Cancel the common factor of $100000$ .

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Step 3.6.1.1.1

Cancel the common factor.

$100000 \frac{y^{\frac{5}{7}}}{100000} = 100000 (- 1 + x)$

Step 3.6.1.1.2

Rewrite the expression.

$y^{\frac{5}{7}} = 100000 (- 1 + x)$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $100000 (- 1 + x)$ .

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Step 3.6.2.1.1

Apply the distributive property.

$y^{\frac{5}{7}} = 100000 \cdot - 1 + 100000 x$

Step 3.6.2.1.2

Multiply $100000$ by $- 1$ .

$y^{\frac{5}{7}} = - 100000 + 100000 x$

Step 3.7

Raise each side of the equation to the power of $\frac{7}{5}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{5}{7}})}^{\frac{7}{5}} = {(- 100000 + 100000 x)}^{\frac{7}{5}}$

Step 3.8

Simplify the left side.

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Step 3.8.1

Simplify ${(y^{\frac{5}{7}})}^{\frac{7}{5}}$ .

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Step 3.8.1.1

Multiply the exponents in ${(y^{\frac{5}{7}})}^{\frac{7}{5}}$ .

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Step 3.8.1.1.1

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

$y^{\frac{5}{7} \cdot \frac{7}{5}} = {(- 100000 + 100000 x)}^{\frac{7}{5}}$

Step 3.8.1.1.2

Cancel the common factor of $5$ .

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Step 3.8.1.1.2.1

Cancel the common factor.

$y^{\frac{5}{7} \cdot \frac{7}{5}} = {(- 100000 + 100000 x)}^{\frac{7}{5}}$

Step 3.8.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{7} \cdot 7} = {(- 100000 + 100000 x)}^{\frac{7}{5}}$

Step 3.8.1.1.3

Cancel the common factor of $7$ .

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Step 3.8.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{7} \cdot 7} = {(- 100000 + 100000 x)}^{\frac{7}{5}}$

Step 3.8.1.1.3.2

Rewrite the expression.

$y^{1} = {(- 100000 + 100000 x)}^{\frac{7}{5}}$

Step 3.8.1.2

Simplify.

$y = {(- 100000 + 100000 x)}^{\frac{7}{5}}$

Step 4

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

$f^{- 1} (x) = {(- 100000 + 100000 x)}^{\frac{7}{5}}$

Step 5

Verify if $f^{- 1} (x) = {(- 100000 + 100000 x)}^{\frac{7}{5}}$ is the inverse of $f (x) = {(\frac{\sqrt[7]{x}}{10})}^{5} + 1$ .

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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 $f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {(- 100000 + 100000 ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1))}^{\frac{7}{5}}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Apply the product rule to $\frac{\sqrt[7]{x}}{10}$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {(- 100000 + 100000 (\frac{{\sqrt[7]{x}}^{5}}{10^{5}} + 1))}^{\frac{7}{5}}$

Step 5.2.3.1.2

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {(- 100000 + 100000 (\frac{\sqrt[7]{x^{5}}}{10^{5}} + 1))}^{\frac{7}{5}}$

Step 5.2.3.1.3

Raise $10$ to the power of $5$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {(- 100000 + 100000 (\frac{\sqrt[7]{x^{5}}}{100000} + 1))}^{\frac{7}{5}}$

Step 5.2.3.2

Apply the distributive property.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {(- 100000 + 100000 (\frac{\sqrt[7]{x^{5}}}{100000}) + 100000 \cdot 1)}^{\frac{7}{5}}$

Step 5.2.3.3

Cancel the common factor of $100000$ .

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Step 5.2.3.3.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {(- 100000 + 100000 (\frac{\sqrt[7]{x^{5}}}{100000}) + 100000 \cdot 1)}^{\frac{7}{5}}$

Step 5.2.3.3.2

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {(- 100000 + \sqrt[7]{x^{5}} + 100000 \cdot 1)}^{\frac{7}{5}}$

Step 5.2.3.4

Multiply $100000$ by $1$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {(- 100000 + \sqrt[7]{x^{5}} + 100000)}^{\frac{7}{5}}$

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Combine the opposite terms in $- 100000 + \sqrt[7]{x^{5}} + 100000$ .

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Step 5.2.4.1.1

Add $- 100000$ and $100000$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {(0 + \sqrt[7]{x^{5}})}^{\frac{7}{5}}$

Step 5.2.4.1.2

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {\sqrt[7]{x^{5}}}^{\frac{7}{5}}$

Step 5.2.4.2

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

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Step 5.2.4.2.1

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = {(x^{\frac{5}{7}})}^{\frac{7}{5}}$

Step 5.2.4.2.2

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

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = x^{\frac{5}{7} \cdot \frac{7}{5}}$

Step 5.2.4.2.3

Multiply $\frac{5}{7}$ by $\frac{7}{5}$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = x^{\frac{5 \cdot 7}{7 \cdot 5}}$

Step 5.2.4.2.4

Multiply $5$ by $7$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = x^{\frac{35}{7 \cdot 5}}$

Step 5.2.4.2.5

Multiply $7$ by $5$ .

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = x^{\frac{35}{35}}$

Step 5.2.4.2.6

Cancel the common factor of $35$ .

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Step 5.2.4.2.6.1

Cancel the common factor.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = x^{\frac{35}{35}}$

Step 5.2.4.2.6.2

Rewrite the expression.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = x$

Step 5.2.4.2.7

Simplify.

$f^{- 1} ({(\frac{\sqrt[7]{x}}{10})}^{5} + 1) = x$

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 $f ({(- 100000 + 100000 x)}^{\frac{7}{5}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = {(\frac{\sqrt[7]{{(- 100000 + 100000 x)}^{\frac{7}{5}}}}{10})}^{5} + 1$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the numerator.

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Step 5.3.3.1.1

Rewrite ${(- 100000 + 100000 x)}^{\frac{7}{5}}$ as ${({(- 100000 + 100000 x)}^{\frac{1}{5}})}^{7}$ .

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = {(\frac{\sqrt[7]{{({(- 100000 + 100000 x)}^{\frac{1}{5}})}^{7}}}{10})}^{5} + 1$

Step 5.3.3.1.2

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

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = {(\frac{{(- 100000 + 100000 x)}^{\frac{1}{5}}}{10})}^{5} + 1$

Step 5.3.3.2

Apply the product rule to $\frac{{(- 100000 + 100000 x)}^{\frac{1}{5}}}{10}$ .

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = \frac{{({(- 100000 + 100000 x)}^{\frac{1}{5}})}^{5}}{10^{5}} + 1$

Step 5.3.3.3

Simplify the numerator.

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Step 5.3.3.3.1

Multiply the exponents in ${({(- 100000 + 100000 x)}^{\frac{1}{5}})}^{5}$ .

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Step 5.3.3.3.1.1

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

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = \frac{{(- 100000 + 100000 x)}^{\frac{1}{5} \cdot 5}}{10^{5}} + 1$

Step 5.3.3.3.1.2

Cancel the common factor of $5$ .

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Step 5.3.3.3.1.2.1

Cancel the common factor.

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = \frac{{(- 100000 + 100000 x)}^{\frac{1}{5} \cdot 5}}{10^{5}} + 1$

Step 5.3.3.3.1.2.2

Rewrite the expression.

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = \frac{- 100000 + 100000 x}{10^{5}} + 1$

Step 5.3.3.3.2

Simplify.

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = \frac{- 100000 + 100000 x}{10^{5}} + 1$

Step 5.3.3.3.3

Factor $100000$ out of $- 100000 + 100000 x$ .

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Step 5.3.3.3.3.1

Factor $100000$ out of $- 100000$ .

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = \frac{100000 \cdot - 1 + 100000 x}{10^{5}} + 1$

Step 5.3.3.3.3.2

Factor $100000$ out of $100000 \cdot - 1 + 100000 x$ .

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = \frac{100000 (- 1 + x)}{10^{5}} + 1$

Step 5.3.3.4

Raise $10$ to the power of $5$ .

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = \frac{100000 (- 1 + x)}{100000} + 1$

Step 5.3.3.5

Cancel the common factor of $100000$ .

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Step 5.3.3.5.1

Cancel the common factor.

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = \frac{100000 (- 1 + x)}{100000} + 1$

Step 5.3.3.5.2

Divide $- 1 + x$ by $1$ .

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = - 1 + x + 1$

Step 5.3.4

Combine the opposite terms in $- 1 + x + 1$ .

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Step 5.3.4.1

Add $- 1$ and $1$ .

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f ({(- 100000 + 100000 x)}^{\frac{7}{5}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = {(- 100000 + 100000 x)}^{\frac{7}{5}}$ is the inverse of $f (x) = {(\frac{\sqrt[7]{x}}{10})}^{5} + 1$ .

$f^{- 1} (x) = {(- 100000 + 100000 x)}^{\frac{7}{5}}$