Find the Inverse f(x)=((x^7-2)/3)^(1/5)

$f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$

Step 1

Write $f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$ as an equation.

$y = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$

Step 2

Interchange the variables.

$x = {(\frac{y^{7} - 2}{3})}^{\frac{1}{5}}$

Step 3

Solve for $y$ .

Rewrite the equation as ${(\frac{y^{7} - 2}{3})}^{\frac{1}{5}} = x$ .

${(\frac{y^{7} - 2}{3})}^{\frac{1}{5}} = x$

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${({(\frac{y^{7} - 2}{3})}^{\frac{1}{5}})}^{5} = x^{5}$

Simplify the left side.

Simplify ${({(\frac{y^{7} - 2}{3})}^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${({(\frac{y^{7} - 2}{3})}^{\frac{1}{5}})}^{5}$ .

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

${(\frac{y^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} = x^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(\frac{y^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} = x^{5}$

Rewrite the expression.

${(\frac{y^{7} - 2}{3})}^{1} = x^{5}$

Split the fraction $\frac{y^{7} - 2}{3}$ into two fractions.

${(\frac{y^{7}}{3} + \frac{- 2}{3})}^{1} = x^{5}$

Move the negative in front of the fraction.

${(\frac{y^{7}}{3} - \frac{2}{3})}^{1} = x^{5}$

Simplify.

$\frac{y^{7}}{3} - \frac{2}{3} = x^{5}$

Solve for $y$ .

Add $\frac{2}{3}$ to both sides of the equation.

$\frac{y^{7}}{3} = x^{5} + \frac{2}{3}$

Multiply both sides of the equation by $3$ .

$3 \frac{y^{7}}{3} = 3 (x^{5} + \frac{2}{3})$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$3 \frac{y^{7}}{3} = 3 (x^{5} + \frac{2}{3})$

Rewrite the expression.

$y^{7} = 3 (x^{5} + \frac{2}{3})$

Simplify the right side.

Simplify $3 (x^{5} + \frac{2}{3})$ .

Apply the distributive property.

$y^{7} = 3 x^{5} + 3 (\frac{2}{3})$

Cancel the common factor of $3$ .

Cancel the common factor.

$y^{7} = 3 x^{5} + 3 (\frac{2}{3})$

Rewrite the expression.

$y^{7} = 3 x^{5} + 2$

Take the 7th root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[7]{3 x^{5} + 2}$

Step 4

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

$f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$

Step 5

Verify if $f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$ is the inverse of $f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$ .

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 {({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}})}^{5} + 2}$

Multiply the exponents in ${({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}})}^{5}$ .

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

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 {(\frac{x^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} + 2}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 {(\frac{x^{7} - 2}{3})}^{\frac{1}{5} \cdot 5} + 2}$

Rewrite the expression.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 (\frac{x^{7} - 2}{3}) + 2}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{3 (\frac{x^{7} - 2}{3}) + 2}$

Rewrite the expression.

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{x^{7} - 2 + 2}$

Simplify by adding numbers.

Add $- 2$ and $2$ .

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{x^{7} + 0}$

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

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = \sqrt[7]{x^{7}}$

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

$f^{- 1} ({(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}) = x$

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (\sqrt[7]{3 x^{5} + 2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{(\sqrt[7]{3 x^{5} + 2})}^{7} - 2}{3})}^{\frac{1}{5}}$

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{3 x^{5} + 2}$ as ${(3 x^{5} + 2)}^{\frac{1}{7}}$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{({(3 x^{5} + 2)}^{\frac{1}{7}})}^{7} - 2}{3})}^{\frac{1}{5}}$

Multiply the exponents in ${({(3 x^{5} + 2)}^{\frac{1}{7}})}^{7}$ .

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

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{(3 x^{5} + 2)}^{\frac{1}{7} \cdot 7} - 2}{3})}^{\frac{1}{5}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{{(3 x^{5} + 2)}^{\frac{1}{7} \cdot 7} - 2}{3})}^{\frac{1}{5}}$

Rewrite the expression.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{(3 x^{5} + 2) - 2}{3})}^{\frac{1}{5}}$

Simplify.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5} + 2 - 2}{3})}^{\frac{1}{5}}$

Subtract $2$ from $2$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5} + 0}{3})}^{\frac{1}{5}}$

Add $3 x^{5}$ and $0$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5}}{3})}^{\frac{1}{5}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\sqrt[7]{3 x^{5} + 2}) = {(\frac{3 x^{5}}{3})}^{\frac{1}{5}}$

Divide $x^{5}$ by $1$ .

$f (\sqrt[7]{3 x^{5} + 2}) = {(x^{5})}^{\frac{1}{5}}$

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

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

$f (\sqrt[7]{3 x^{5} + 2}) = x^{5 (\frac{1}{5})}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\sqrt[7]{3 x^{5} + 2}) = x^{5 (\frac{1}{5})}$

Rewrite the expression.

$f (\sqrt[7]{3 x^{5} + 2}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$ is the inverse of $f (x) = {(\frac{x^{7} - 2}{3})}^{\frac{1}{5}}$ .

$f^{- 1} (x) = \sqrt[7]{3 x^{5} + 2}$

Find the Inverse f(x)=((x^7-2)/3)^(1/5)

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