Precalculus Examples

$f (x) = \frac{x^{7}}{8} + 8$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = \frac{x^{7}}{8} + 8$ as an equation.

$y = \frac{x^{7}}{8} + 8$

Step 2

Interchange the variables.

$x = \frac{y^{7}}{8} + 8$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{y^{7}}{8} + 8 = x$ .

$\frac{y^{7}}{8} + 8 = x$

Step 3.2

Subtract $8$ from both sides of the equation.

$\frac{y^{7}}{8} = x - 8$

Step 3.3

Multiply both sides of the equation by $8$ .

$8 \frac{y^{7}}{8} = 8 (x - 8)$

Step 3.4

Simplify both sides of the equation.

Tap for more steps...

Step 3.4.1

Simplify the left side.

Tap for more steps...

Step 3.4.1.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 3.4.1.1.1

Cancel the common factor.

$8 \frac{y^{7}}{8} = 8 (x - 8)$

Step 3.4.1.1.2

Rewrite the expression.

$y^{7} = 8 (x - 8)$

Step 3.4.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.1

Simplify $8 (x - 8)$ .

Tap for more steps...

Step 3.4.2.1.1

Apply the distributive property.

$y^{7} = 8 x + 8 \cdot - 8$

Step 3.4.2.1.2

Multiply $8$ by $- 8$ .

$y^{7} = 8 x - 64$

Step 3.5

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

$y = \sqrt[7]{8 x - 64}$

Step 3.6

Factor $8$ out of $8 x - 64$ .

Tap for more steps...

Step 3.6.1

Factor $8$ out of $8 x$ .

$y = \sqrt[7]{8 (x) - 64}$

Step 3.6.2

Factor $8$ out of $- 64$ .

$y = \sqrt[7]{8 x + 8 \cdot - 8}$

Step 3.6.3

Factor $8$ out of $8 x + 8 \cdot - 8$ .

$y = \sqrt[7]{8 (x - 8)}$

Step 4

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

$f^{- 1} (x) = \sqrt[7]{8 (x - 8)}$

Step 5

Verify if $f^{- 1} (x) = \sqrt[7]{8 (x - 8)}$ is the inverse of $f (x) = \frac{x^{7}}{8} + 8$ .

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 $f^{- 1} (\frac{x^{7}}{8} + 8)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x^{7}}{8} + 8) = \sqrt[7]{8 ((\frac{x^{7}}{8} + 8) - 8)}$

Step 5.2.3

Simplify by subtracting numbers.

Tap for more steps...

Step 5.2.3.1

Subtract $8$ from $8$ .

$f^{- 1} (\frac{x^{7}}{8} + 8) = \sqrt[7]{8 (\frac{x^{7}}{8} + 0)}$

Step 5.2.3.2

Add $\frac{x^{7}}{8}$ and $0$ .

$f^{- 1} (\frac{x^{7}}{8} + 8) = \sqrt[7]{8 (\frac{x^{7}}{8})}$

Step 5.2.4

Combine $8$ and $\frac{x^{7}}{8}$ .

$f^{- 1} (\frac{x^{7}}{8} + 8) = \sqrt[7]{\frac{8 x^{7}}{8}}$

Step 5.2.5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.5.1

Reduce the expression $\frac{8 x^{7}}{8}$ by cancelling the common factors.

Tap for more steps...

Step 5.2.5.1.1

Cancel the common factor.

$f^{- 1} (\frac{x^{7}}{8} + 8) = \sqrt[7]{\frac{8 x^{7}}{8}}$

Step 5.2.5.1.2

Rewrite the expression.

$f^{- 1} (\frac{x^{7}}{8} + 8) = \sqrt[7]{\frac{x^{7}}{1}}$

Step 5.2.5.2

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

$f^{- 1} (\frac{x^{7}}{8} + 8) = \sqrt[7]{x^{7}}$

Step 5.2.6

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

$f^{- 1} (\frac{x^{7}}{8} + 8) = x$

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 $f (\sqrt[7]{8 (x - 8)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[7]{8 (x - 8)}) = \frac{{(\sqrt[7]{8 (x - 8)})}^{7}}{8} + 8$

Step 5.3.3

Simplify each term.

Tap for more steps...

Step 5.3.3.1

Simplify the numerator.

Tap for more steps...

Step 5.3.3.1.1

Rewrite ${\sqrt[7]{8 (x - 8)}}^{7}$ as $8 (x - 8)$ .

Tap for more steps...

Step 5.3.3.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{8 (x - 8)}$ as ${(8 (x - 8))}^{\frac{1}{7}}$ .

$f (\sqrt[7]{8 (x - 8)}) = \frac{{({(8 (x - 8))}^{\frac{1}{7}})}^{7}}{8} + 8$

Step 5.3.3.1.1.2

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

$f (\sqrt[7]{8 (x - 8)}) = \frac{{(8 (x - 8))}^{\frac{1}{7} \cdot 7}}{8} + 8$

Step 5.3.3.1.1.3

Combine $\frac{1}{7}$ and $7$ .

$f (\sqrt[7]{8 (x - 8)}) = \frac{{(8 (x - 8))}^{\frac{7}{7}}}{8} + 8$

Step 5.3.3.1.1.4

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.3.3.1.1.4.1

Cancel the common factor.

$f (\sqrt[7]{8 (x - 8)}) = \frac{{(8 (x - 8))}^{\frac{7}{7}}}{8} + 8$

Step 5.3.3.1.1.4.2

Rewrite the expression.

$f (\sqrt[7]{8 (x - 8)}) = \frac{8 (x - 8)}{8} + 8$

Step 5.3.3.1.1.5

Simplify.

$f (\sqrt[7]{8 (x - 8)}) = \frac{8 (x - 8)}{8} + 8$

Step 5.3.3.1.2

Apply the distributive property.

$f (\sqrt[7]{8 (x - 8)}) = \frac{8 x + 8 \cdot - 8}{8} + 8$

Step 5.3.3.1.3

Multiply $8$ by $- 8$ .

$f (\sqrt[7]{8 (x - 8)}) = \frac{8 x - 64}{8} + 8$

Step 5.3.3.1.4

Factor $8$ out of $8 x - 64$ .

Tap for more steps...

Step 5.3.3.1.4.1

Factor $8$ out of $8 x$ .

$f (\sqrt[7]{8 (x - 8)}) = \frac{8 (x) - 64}{8} + 8$

Step 5.3.3.1.4.2

Factor $8$ out of $- 64$ .

$f (\sqrt[7]{8 (x - 8)}) = \frac{8 x + 8 \cdot - 8}{8} + 8$

Step 5.3.3.1.4.3

Factor $8$ out of $8 x + 8 \cdot - 8$ .

$f (\sqrt[7]{8 (x - 8)}) = \frac{8 (x - 8)}{8} + 8$

Step 5.3.3.2

Cancel the common factor of $8$ .

Tap for more steps...

Step 5.3.3.2.1

Cancel the common factor.

$f (\sqrt[7]{8 (x - 8)}) = \frac{8 (x - 8)}{8} + 8$

Step 5.3.3.2.2

Divide $x - 8$ by $1$ .

$f (\sqrt[7]{8 (x - 8)}) = x - 8 + 8$

Step 5.3.4

Combine the opposite terms in $x - 8 + 8$ .

Tap for more steps...

Step 5.3.4.1

Add $- 8$ and $8$ .

$f (\sqrt[7]{8 (x - 8)}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\sqrt[7]{8 (x - 8)}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[7]{8 (x - 8)}$ is the inverse of $f (x) = \frac{x^{7}}{8} + 8$ .

$f^{- 1} (x) = \sqrt[7]{8 (x - 8)}$