Algebra Examples

$f (x) = {(\frac{x - 10}{7})}^{3}$

Step 1

Write $f (x) = {(\frac{x - 10}{7})}^{3}$ as an equation.

$y = {(\frac{x - 10}{7})}^{3}$

Step 2

Interchange the variables.

$x = {(\frac{y - 10}{7})}^{3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as ${(\frac{y - 10}{7})}^{3} = x$ .

${(\frac{y - 10}{7})}^{3} = x$

Step 3.2

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

$\frac{y - 10}{7} = \sqrt[3]{x}$

Step 3.3

Multiply both sides of the equation by $7$ .

$7 \frac{y - 10}{7} = 7 \sqrt[3]{x}$

Step 3.4

Simplify the left side.

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Step 3.4.1

Cancel the common factor of $7$ .

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Step 3.4.1.1

Cancel the common factor.

$7 \frac{y - 10}{7} = 7 \sqrt[3]{x}$

Step 3.4.1.2

Rewrite the expression.

$y - 10 = 7 \sqrt[3]{x}$

Step 3.5

Add $10$ to both sides of the equation.

$y = 7 \sqrt[3]{x} + 10$

Step 4

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

$f^{- 1} (x) = 7 \sqrt[3]{x} + 10$

Step 5

Verify if $f^{- 1} (x) = 7 \sqrt[3]{x} + 10$ is the inverse of $f (x) = {(\frac{x - 10}{7})}^{3}$ .

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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 $f^{- 1} ({(\frac{x - 10}{7})}^{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(\frac{x - 10}{7})}^{3}) = 7 \sqrt[3]{{(\frac{x - 10}{7})}^{3}} + 10$

Step 5.2.3

Remove parentheses.

$f^{- 1} ({(\frac{x - 10}{7})}^{3}) = 7 \sqrt[3]{{(\frac{x - 10}{7})}^{3}} + 10$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

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

$f^{- 1} ({(\frac{x - 10}{7})}^{3}) = 7 (\frac{x - 10}{7}) + 10$

Step 5.2.4.2

Cancel the common factor of $7$ .

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Step 5.2.4.2.1

Cancel the common factor.

$f^{- 1} ({(\frac{x - 10}{7})}^{3}) = 7 (\frac{x - 10}{7}) + 10$

Step 5.2.4.2.2

Rewrite the expression.

$f^{- 1} ({(\frac{x - 10}{7})}^{3}) = x - 10 + 10$

Step 5.2.5

Combine the opposite terms in $x - 10 + 10$ .

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Step 5.2.5.1

Add $- 10$ and $10$ .

$f^{- 1} ({(\frac{x - 10}{7})}^{3}) = x + 0$

Step 5.2.5.2

Add $x$ and $0$ .

$f^{- 1} ({(\frac{x - 10}{7})}^{3}) = x$

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 $f (7 \sqrt[3]{x} + 10)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (7 \sqrt[3]{x} + 10) = {(\frac{(7 \sqrt[3]{x} + 10) - 10}{7})}^{3}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Subtract $10$ from $10$ .

$f (7 \sqrt[3]{x} + 10) = {(\frac{7 \sqrt[3]{x} + 0}{7})}^{3}$

Step 5.3.3.2

Add $7 \sqrt[3]{x}$ and $0$ .

$f (7 \sqrt[3]{x} + 10) = {(\frac{7 \sqrt[3]{x}}{7})}^{3}$

Step 5.3.4

Simplify terms.

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Step 5.3.4.1

Cancel the common factor of $7$ .

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Step 5.3.4.1.1

Cancel the common factor.

$f (7 \sqrt[3]{x} + 10) = {(\frac{7 \sqrt[3]{x}}{7})}^{3}$

Step 5.3.4.1.2

Divide $\sqrt[3]{x}$ by $1$ .

$f (7 \sqrt[3]{x} + 10) = {\sqrt[3]{x}}^{3}$

Step 5.3.4.2

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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Step 5.3.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$f (7 \sqrt[3]{x} + 10) = {(x^{\frac{1}{3}})}^{3}$

Step 5.3.4.2.2

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

$f (7 \sqrt[3]{x} + 10) = x^{\frac{1}{3} \cdot 3}$

Step 5.3.4.2.3

Combine $\frac{1}{3}$ and $3$ .

$f (7 \sqrt[3]{x} + 10) = x^{\frac{3}{3}}$

Step 5.3.4.2.4

Cancel the common factor of $3$ .

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Step 5.3.4.2.4.1

Cancel the common factor.

$f (7 \sqrt[3]{x} + 10) = x^{\frac{3}{3}}$

Step 5.3.4.2.4.2

Rewrite the expression.

$f (7 \sqrt[3]{x} + 10) = x$

Step 5.3.4.2.5

Simplify.

$f (7 \sqrt[3]{x} + 10) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 7 \sqrt[3]{x} + 10$ is the inverse of $f (x) = {(\frac{x - 10}{7})}^{3}$ .

$f^{- 1} (x) = 7 \sqrt[3]{x} + 10$