Find the Inverse f(x)=((x+9)^(1/5))/8

$f (x) = \frac{{(x + 9)}^{\frac{1}{5}}}{8}$

Step 1

Write $f (x) = \frac{{(x + 9)}^{\frac{1}{5}}}{8}$ as an equation.

$y = \frac{{(x + 9)}^{\frac{1}{5}}}{8}$

Step 2

Interchange the variables.

$x = \frac{{(y + 9)}^{\frac{1}{5}}}{8}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{{(y + 9)}^{\frac{1}{5}}}{8} = x$ .

$\frac{{(y + 9)}^{\frac{1}{5}}}{8} = x$

Multiply both sides of the equation by $8$ .

$8 \frac{{(y + 9)}^{\frac{1}{5}}}{8} = 8 x$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$8 \frac{{(y + 9)}^{\frac{1}{5}}}{8} = 8 x$

Rewrite the expression.

${(y + 9)}^{\frac{1}{5}} = 8 x$

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

${({(y + 9)}^{\frac{1}{5}})}^{5} = {(8 x)}^{5}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(y + 9)}^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${({(y + 9)}^{\frac{1}{5}})}^{5}$ .

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

${(y + 9)}^{\frac{1}{5} \cdot 5} = {(8 x)}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(y + 9)}^{\frac{1}{5} \cdot 5} = {(8 x)}^{5}$

Rewrite the expression.

${(y + 9)}^{1} = {(8 x)}^{5}$

Simplify.

$y + 9 = {(8 x)}^{5}$

Simplify the right side.

Simplify ${(8 x)}^{5}$ .

Apply the product rule to $8 x$ .

$y + 9 = 8^{5} x^{5}$

Raise $8$ to the power of $5$ .

$y + 9 = 32768 x^{5}$

Subtract $9$ from both sides of the equation.

$y = 32768 x^{5} - 9$

Step 4

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

$f^{- 1} (x) = 32768 x^{5} - 9$

Step 5

Verify if $f^{- 1} (x) = 32768 x^{5} - 9$ is the inverse of $f (x) = \frac{{(x + 9)}^{\frac{1}{5}}}{8}$ .

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 $f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = 32768 {(\frac{{(x + 9)}^{\frac{1}{5}}}{8})}^{5} - 9$

Simplify each term.

Apply the product rule to $\frac{{(x + 9)}^{\frac{1}{5}}}{8}$ .

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = 32768 (\frac{{({(x + 9)}^{\frac{1}{5}})}^{5}}{8^{5}}) - 9$

Simplify the numerator.

Multiply the exponents in ${({(x + 9)}^{\frac{1}{5}})}^{5}$ .

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

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = 32768 (\frac{{(x + 9)}^{\frac{1}{5} \cdot 5}}{8^{5}}) - 9$

Cancel the common factor of $5$ .

Cancel the common factor.

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = 32768 (\frac{{(x + 9)}^{\frac{1}{5} \cdot 5}}{8^{5}}) - 9$

Rewrite the expression.

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = 32768 (\frac{x + 9}{8^{5}}) - 9$

Simplify.

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = 32768 (\frac{x + 9}{8^{5}}) - 9$

Raise $8$ to the power of $5$ .

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = 32768 (\frac{x + 9}{32768}) - 9$

Cancel the common factor of $32768$ .

Cancel the common factor.

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = 32768 (\frac{x + 9}{32768}) - 9$

Rewrite the expression.

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = x + 9 - 9$

Combine the opposite terms in $x + 9 - 9$ .

Subtract $9$ from $9$ .

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{{(x + 9)}^{\frac{1}{5}}}{8}) = x$

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

Set up the composite result function.

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

Evaluate $f (32768 x^{5} - 9)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (32768 x^{5} - 9) = \frac{{((32768 x^{5} - 9) + 9)}^{\frac{1}{5}}}{8}$

Simplify the numerator.

Combine the opposite terms in $32768 x^{5} - 9 + 9$ .

Add $- 9$ and $9$ .

$f (32768 x^{5} - 9) = \frac{{(32768 x^{5} + 0)}^{\frac{1}{5}}}{8}$

Add $32768 x^{5}$ and $0$ .

$f (32768 x^{5} - 9) = \frac{{(32768 x^{5})}^{\frac{1}{5}}}{8}$

Apply the product rule to $32768 x^{5}$ .

$f (32768 x^{5} - 9) = \frac{32768^{\frac{1}{5}} {(x^{5})}^{\frac{1}{5}}}{8}$

Rewrite $32768$ as $8^{5}$ .

$f (32768 x^{5} - 9) = \frac{{(8^{5})}^{\frac{1}{5}} {(x^{5})}^{\frac{1}{5}}}{8}$

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

$f (32768 x^{5} - 9) = \frac{8^{5 (\frac{1}{5})} {(x^{5})}^{\frac{1}{5}}}{8}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (32768 x^{5} - 9) = \frac{8^{5 (\frac{1}{5})} {(x^{5})}^{\frac{1}{5}}}{8}$

Rewrite the expression.

$f (32768 x^{5} - 9) = \frac{8 {(x^{5})}^{\frac{1}{5}}}{8}$

Evaluate the exponent.

$f (32768 x^{5} - 9) = \frac{8 {(x^{5})}^{\frac{1}{5}}}{8}$

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

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

$f (32768 x^{5} - 9) = \frac{8 x^{5 (\frac{1}{5})}}{8}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (32768 x^{5} - 9) = \frac{8 x^{5 (\frac{1}{5})}}{8}$

Rewrite the expression.

$f (32768 x^{5} - 9) = \frac{8 x}{8}$

Simplify.

$f (32768 x^{5} - 9) = \frac{8 x}{8}$

Cancel the common factor of $8$ .

Cancel the common factor.

$f (32768 x^{5} - 9) = \frac{8 x}{8}$

Divide $x$ by $1$ .

$f (32768 x^{5} - 9) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 32768 x^{5} - 9$ is the inverse of $f (x) = \frac{{(x + 9)}^{\frac{1}{5}}}{8}$ .

$f^{- 1} (x) = 32768 x^{5} - 9$

Find the Inverse f(x)=((x+9)^(1/5))/8

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