Precalculus Examples

$f (x) = \frac{\sqrt[5]{x - 2}}{9}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \frac{\sqrt[5]{x - 2}}{9}$ as an equation.

$y = \frac{\sqrt[5]{x - 2}}{9}$

Step 2

Interchange the variables.

$x = \frac{\sqrt[5]{y - 2}}{9}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{\sqrt[5]{y - 2}}{9} = x$ .

$\frac{\sqrt[5]{y - 2}}{9} = x$

Step 3.2

Multiply both sides of the equation by $9$ .

$9 \frac{\sqrt[5]{y - 2}}{9} = 9 x$

Step 3.3

Simplify the left side.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $9$ .

Tap for more steps...

Step 3.3.1.1

Cancel the common factor.

$9 \frac{\sqrt[5]{y - 2}}{9} = 9 x$

Step 3.3.1.2

Rewrite the expression.

$\sqrt[5]{y - 2} = 9 x$

Step 3.4

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{y - 2}}^{5} = {(9 x)}^{5}$

Step 3.5

Simplify each side of the equation.

Tap for more steps...

Step 3.5.1

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

${({(y - 2)}^{\frac{1}{5}})}^{5} = {(9 x)}^{5}$

Step 3.5.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.1

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

Tap for more steps...

Step 3.5.2.1.1

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

Tap for more steps...

Step 3.5.2.1.1.1

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

${(y - 2)}^{\frac{1}{5} \cdot 5} = {(9 x)}^{5}$

Step 3.5.2.1.1.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.5.2.1.1.2.1

Cancel the common factor.

${(y - 2)}^{\frac{1}{5} \cdot 5} = {(9 x)}^{5}$

Step 3.5.2.1.1.2.2

Rewrite the expression.

${(y - 2)}^{1} = {(9 x)}^{5}$

Step 3.5.2.1.2

Simplify.

$y - 2 = {(9 x)}^{5}$

Step 3.5.3

Simplify the right side.

Tap for more steps...

Step 3.5.3.1

Simplify ${(9 x)}^{5}$ .

Tap for more steps...

Step 3.5.3.1.1

Apply the product rule to $9 x$ .

$y - 2 = 9^{5} x^{5}$

Step 3.5.3.1.2

Raise $9$ to the power of $5$ .

$y - 2 = 59049 x^{5}$

Step 3.6

Add $2$ to both sides of the equation.

$y = 59049 x^{5} + 2$

Step 4

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

$f^{- 1} (x) = 59049 x^{5} + 2$

Step 5

Verify if $f^{- 1} (x) = 59049 x^{5} + 2$ is the inverse of $f (x) = \frac{\sqrt[5]{x - 2}}{9}$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{\sqrt[5]{x - 2}}{9})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = 59049 {(\frac{\sqrt[5]{x - 2}}{9})}^{5} + 2$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Apply the product rule to $\frac{\sqrt[5]{x - 2}}{9}$ .

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = 59049 (\frac{{\sqrt[5]{x - 2}}^{5}}{9^{5}}) + 2$

Step 5.2.3.2

Rewrite ${\sqrt[5]{x - 2}}^{5}$ as $x - 2$ .

Tap for more steps...

Step 5.2.3.2.1

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

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = 59049 (\frac{{({(x - 2)}^{\frac{1}{5}})}^{5}}{9^{5}}) + 2$

Step 5.2.3.2.2

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

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = 59049 (\frac{{(x - 2)}^{\frac{1}{5} \cdot 5}}{9^{5}}) + 2$

Step 5.2.3.2.3

Combine $\frac{1}{5}$ and $5$ .

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = 59049 (\frac{{(x - 2)}^{\frac{5}{5}}}{9^{5}}) + 2$

Step 5.2.3.2.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.3.2.4.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = 59049 (\frac{{(x - 2)}^{\frac{5}{5}}}{9^{5}}) + 2$

Step 5.2.3.2.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = 59049 (\frac{x - 2}{9^{5}}) + 2$

Step 5.2.3.2.5

Simplify.

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = 59049 (\frac{x - 2}{9^{5}}) + 2$

Step 5.2.3.3

Raise $9$ to the power of $5$ .

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = 59049 (\frac{x - 2}{59049}) + 2$

Step 5.2.3.4

Cancel the common factor of $59049$ .

Tap for more steps...

Step 5.2.3.4.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = 59049 (\frac{x - 2}{59049}) + 2$

Step 5.2.3.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = x - 2 + 2$

Step 5.2.4

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

Tap for more steps...

Step 5.2.4.1

Add $- 2$ and $2$ .

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{\sqrt[5]{x - 2}}{9}) = x$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (59049 x^{5} + 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (59049 x^{5} + 2) = \frac{\sqrt[5]{(59049 x^{5} + 2) - 2}}{9}$

Step 5.3.3

Simplify the numerator.

Tap for more steps...

Step 5.3.3.1

Subtract $2$ from $2$ .

$f (59049 x^{5} + 2) = \frac{\sqrt[5]{59049 x^{5} + 0}}{9}$

Step 5.3.3.2

Add $59049 x^{5}$ and $0$ .

$f (59049 x^{5} + 2) = \frac{\sqrt[5]{59049 x^{5}}}{9}$

Step 5.3.3.3

Rewrite $59049 x^{5}$ as ${(9 x)}^{5}$ .

$f (59049 x^{5} + 2) = \frac{\sqrt[5]{{(9 x)}^{5}}}{9}$

Step 5.3.3.4

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

$f (59049 x^{5} + 2) = \frac{9 x}{9}$

Step 5.3.4

Cancel the common factor of $9$ .

Tap for more steps...

Step 5.3.4.1

Cancel the common factor.

$f (59049 x^{5} + 2) = \frac{9 x}{9}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (59049 x^{5} + 2) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 59049 x^{5} + 2$ is the inverse of $f (x) = \frac{\sqrt[5]{x - 2}}{9}$ .

$f^{- 1} (x) = 59049 x^{5} + 2$