Algebra Examples

$f (x) = {(\frac{x + 2}{7})}^{\frac{1}{7}}$

Step 1

Write $f (x) = {(\frac{x + 2}{7})}^{\frac{1}{7}}$ as an equation.

$y = {(\frac{x + 2}{7})}^{\frac{1}{7}}$

Step 2

Interchange the variables.

$x = {(\frac{y + 2}{7})}^{\frac{1}{7}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as ${(\frac{y + 2}{7})}^{\frac{1}{7}} = x$ .

${(\frac{y + 2}{7})}^{\frac{1}{7}} = x$

Step 3.2

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

${({(\frac{y + 2}{7})}^{\frac{1}{7}})}^{7} = x^{7}$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify ${({(\frac{y + 2}{7})}^{\frac{1}{7}})}^{7}$ .

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Step 3.3.1.1

Multiply the exponents in ${({(\frac{y + 2}{7})}^{\frac{1}{7}})}^{7}$ .

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Step 3.3.1.1.1

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

${(\frac{y + 2}{7})}^{\frac{1}{7} \cdot 7} = x^{7}$

Step 3.3.1.1.2

Cancel the common factor of $7$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

${(\frac{y + 2}{7})}^{\frac{1}{7} \cdot 7} = x^{7}$

Step 3.3.1.1.2.2

Rewrite the expression.

${(\frac{y + 2}{7})}^{1} = x^{7}$

Step 3.3.1.2

Split the fraction $\frac{y + 2}{7}$ into two fractions.

${(\frac{y}{7} + \frac{2}{7})}^{1} = x^{7}$

Step 3.3.1.3

Simplify.

$\frac{y}{7} + \frac{2}{7} = x^{7}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $\frac{2}{7}$ from both sides of the equation.

$\frac{y}{7} = x^{7} - \frac{2}{7}$

Step 3.4.2

Multiply both sides of the equation by $7$ .

$7 \frac{y}{7} = 7 (x^{7} - \frac{2}{7})$

Step 3.4.3

Simplify both sides of the equation.

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Step 3.4.3.1

Simplify the left side.

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Step 3.4.3.1.1

Cancel the common factor of $7$ .

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Step 3.4.3.1.1.1

Cancel the common factor.

$7 \frac{y}{7} = 7 (x^{7} - \frac{2}{7})$

Step 3.4.3.1.1.2

Rewrite the expression.

$y = 7 (x^{7} - \frac{2}{7})$

Step 3.4.3.2

Simplify the right side.

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Step 3.4.3.2.1

Simplify $7 (x^{7} - \frac{2}{7})$ .

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Step 3.4.3.2.1.1

Apply the distributive property.

$y = 7 x^{7} + 7 (- \frac{2}{7})$

Step 3.4.3.2.1.2

Cancel the common factor of $7$ .

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Step 3.4.3.2.1.2.1

Move the leading negative in $- \frac{2}{7}$ into the numerator.

$y = 7 x^{7} + 7 (\frac{- 2}{7})$

Step 3.4.3.2.1.2.2

Cancel the common factor.

$y = 7 x^{7} + 7 (\frac{- 2}{7})$

Step 3.4.3.2.1.2.3

Rewrite the expression.

$y = 7 x^{7} - 2$

Step 4

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

$f^{- 1} (x) = 7 x^{7} - 2$

Step 5

Verify if $f^{- 1} (x) = 7 x^{7} - 2$ is the inverse of $f (x) = {(\frac{x + 2}{7})}^{\frac{1}{7}}$ .

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Multiply the exponents in ${({(\frac{x + 2}{7})}^{\frac{1}{7}})}^{7}$ .

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Step 5.2.3.1.1

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

$f^{- 1} ({(\frac{x + 2}{7})}^{\frac{1}{7}}) = 7 {(\frac{x + 2}{7})}^{\frac{1}{7} \cdot 7} - 2$

Step 5.2.3.1.2

Cancel the common factor of $7$ .

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Step 5.2.3.1.2.1

Cancel the common factor.

$f^{- 1} ({(\frac{x + 2}{7})}^{\frac{1}{7}}) = 7 {(\frac{x + 2}{7})}^{\frac{1}{7} \cdot 7} - 2$

Step 5.2.3.1.2.2

Rewrite the expression.

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

Step 5.2.3.2

Simplify.

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

Step 5.2.3.3

Cancel the common factor of $7$ .

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Step 5.2.3.3.1

Cancel the common factor.

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

Step 5.2.3.3.2

Rewrite the expression.

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

Step 5.2.4

Combine the opposite terms in $x + 2 - 2$ .

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Step 5.2.4.1

Subtract $2$ from $2$ .

$f^{- 1} ({(\frac{x + 2}{7})}^{\frac{1}{7}}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

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

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 $f (7 x^{7} - 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Add $- 2$ and $2$ .

$f (7 x^{7} - 2) = {(\frac{7 x^{7} + 0}{7})}^{\frac{1}{7}}$

Step 5.3.3.2

Add $7 x^{7}$ and $0$ .

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

Step 5.3.4

Reduce the expression by cancelling the common factors.

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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.

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

Step 5.3.4.1.2

Divide $x^{7}$ by $1$ .

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

Step 5.3.4.2

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

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Step 5.3.4.2.1

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

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

Step 5.3.4.2.2

Cancel the common factor of $7$ .

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Step 5.3.4.2.2.1

Cancel the common factor.

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

Step 5.3.4.2.2.2

Rewrite the expression.

$f (7 x^{7} - 2) = x$

Step 5.4

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

$f^{- 1} (x) = 7 x^{7} - 2$