Calculus Examples

$f (x) = 9 \sqrt[7]{8 x - 7}$

Step 1

Write $f (x) = 9 \sqrt[7]{8 x - 7}$ as an equation.

$y = 9 \sqrt[7]{8 x - 7}$

Step 2

Interchange the variables.

$x = 9 \sqrt[7]{8 y - 7}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $9 \sqrt[7]{8 y - 7} = x$ .

$9 \sqrt[7]{8 y - 7} = x$

Step 3.2

Divide each term in $9 \sqrt[7]{8 y - 7} = x$ by $9$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $9 \sqrt[7]{8 y - 7} = x$ by $9$ .

$\frac{9 \sqrt[7]{8 y - 7}}{9} = \frac{x}{9}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $9$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{9 \sqrt[7]{8 y - 7}}{9} = \frac{x}{9}$

Step 3.2.2.1.2

Divide $\sqrt[7]{8 y - 7}$ by $1$ .

$\sqrt[7]{8 y - 7} = \frac{x}{9}$

Step 3.3

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $7$ .

${\sqrt[7]{8 y - 7}}^{7} = {(\frac{x}{9})}^{7}$

Step 3.4

Simplify each side of the equation.

Tap for more steps...

Step 3.4.1

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

${({(8 y - 7)}^{\frac{1}{7}})}^{7} = {(\frac{x}{9})}^{7}$

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Simplify ${({(8 y - 7)}^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 3.4.2.1.1

Multiply the exponents in ${({(8 y - 7)}^{\frac{1}{7}})}^{7}$ .

Tap for more steps...

Step 3.4.2.1.1.1

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

${(8 y - 7)}^{\frac{1}{7} \cdot 7} = {(\frac{x}{9})}^{7}$

Step 3.4.2.1.1.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 3.4.2.1.1.2.1

Cancel the common factor.

${(8 y - 7)}^{\frac{1}{7} \cdot 7} = {(\frac{x}{9})}^{7}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

${(8 y - 7)}^{1} = {(\frac{x}{9})}^{7}$

Step 3.4.2.1.2

Simplify.

$8 y - 7 = {(\frac{x}{9})}^{7}$

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Simplify ${(\frac{x}{9})}^{7}$ .

Tap for more steps...

Step 3.4.3.1.1

Apply the product rule to $\frac{x}{9}$ .

$8 y - 7 = \frac{x^{7}}{9^{7}}$

Step 3.4.3.1.2

Raise $9$ to the power of $7$ .

$8 y - 7 = \frac{x^{7}}{4782969}$

Step 3.5

Solve for $y$ .

Tap for more steps...

Step 3.5.1

Add $7$ to both sides of the equation.

$8 y = \frac{x^{7}}{4782969} + 7$

Step 3.5.2

Divide each term in $8 y = \frac{x^{7}}{4782969} + 7$ by $8$ and simplify.

Tap for more steps...

Step 3.5.2.1

Divide each term in $8 y = \frac{x^{7}}{4782969} + 7$ by $8$ .

$\frac{8 y}{8} = \frac{\frac{x^{7}}{4782969}}{8} + \frac{7}{8}$

Step 3.5.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{\frac{x^{7}}{4782969}}{8} + \frac{7}{8}$

Step 3.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{x^{7}}{4782969}}{8} + \frac{7}{8}$

Step 3.5.2.3

Simplify the right side.

Tap for more steps...

Step 3.5.2.3.1

Simplify each term.

Tap for more steps...

Step 3.5.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x^{7}}{4782969} \cdot \frac{1}{8} + \frac{7}{8}$

Step 3.5.2.3.1.2

Combine.

$y = \frac{x^{7} \cdot 1}{4782969 \cdot 8} + \frac{7}{8}$

Step 3.5.2.3.1.3

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

$y = \frac{x^{7}}{4782969 \cdot 8} + \frac{7}{8}$

Step 3.5.2.3.1.4

Multiply $4782969$ by $8$ .

$y = \frac{x^{7}}{38263752} + \frac{7}{8}$

Step 4

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

$f^{- 1} (x) = \frac{x^{7}}{38263752} + \frac{7}{8}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{7}}{38263752} + \frac{7}{8}$ is the inverse of $f (x) = 9 \sqrt[7]{8 x - 7}$ .

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 $f^{- 1} (9 \sqrt[7]{8 x - 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{{(9 \sqrt[7]{8 x - 7})}^{7}}{38263752} + \frac{7}{8}$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1.1

Apply the product rule to $9 \sqrt[7]{8 x - 7}$ .

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{9^{7} {\sqrt[7]{8 x - 7}}^{7}}{38263752} + \frac{7}{8}$

Step 5.2.3.1.2

Raise $9$ to the power of $7$ .

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{4782969 {\sqrt[7]{8 x - 7}}^{7}}{38263752} + \frac{7}{8}$

Step 5.2.3.1.3

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

Tap for more steps...

Step 5.2.3.1.3.1

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

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{4782969 {({(8 x - 7)}^{\frac{1}{7}})}^{7}}{38263752} + \frac{7}{8}$

Step 5.2.3.1.3.2

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

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{4782969 {(8 x - 7)}^{\frac{1}{7} \cdot 7}}{38263752} + \frac{7}{8}$

Step 5.2.3.1.3.3

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

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{4782969 {(8 x - 7)}^{\frac{7}{7}}}{38263752} + \frac{7}{8}$

Step 5.2.3.1.3.4

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.2.3.1.3.4.1

Cancel the common factor.

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{4782969 {(8 x - 7)}^{\frac{7}{7}}}{38263752} + \frac{7}{8}$

Step 5.2.3.1.3.4.2

Rewrite the expression.

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{4782969 (8 x - 7)}{38263752} + \frac{7}{8}$

Step 5.2.3.1.3.5

Simplify.

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{4782969 (8 x - 7)}{38263752} + \frac{7}{8}$

Step 5.2.3.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.2.1

Factor $4782969$ out of $38263752$ .

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{4782969 (8 x - 7)}{4782969 \cdot 8} + \frac{7}{8}$

Step 5.2.3.2.2

Cancel the common factor.

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{4782969 (8 x - 7)}{4782969 \cdot 8} + \frac{7}{8}$

Step 5.2.3.2.3

Rewrite the expression.

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{8 x - 7}{8} + \frac{7}{8}$

Step 5.2.4

Simplify terms.

Tap for more steps...

Step 5.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{8 x - 7 + 7}{8}$

Step 5.2.4.2

Combine the opposite terms in $8 x - 7 + 7$ .

Tap for more steps...

Step 5.2.4.2.1

Add $- 7$ and $7$ .

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{8 x + 0}{8}$

Step 5.2.4.2.2

Add $8 x$ and $0$ .

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{8 x}{8}$

Step 5.2.4.3

Cancel the common factor of $8$ .

Tap for more steps...

Step 5.2.4.3.1

Cancel the common factor.

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = \frac{8 x}{8}$

Step 5.2.4.3.2

Divide $x$ by $1$ .

$f^{- 1} (9 \sqrt[7]{8 x - 7}) = x$

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 $f (\frac{x^{7}}{38263752} + \frac{7}{8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{8 (\frac{x^{7}}{38263752} + \frac{7}{8}) - 7}$

Step 5.3.3

Apply the distributive property.

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{8 (\frac{x^{7}}{38263752}) + 8 (\frac{7}{8}) - 7}$

Step 5.3.4

Cancel the common factor of $8$ .

Tap for more steps...

Step 5.3.4.1

Factor $8$ out of $38263752$ .

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{8 (\frac{x^{7}}{8 (4782969)}) + 8 (\frac{7}{8}) - 7}$

Step 5.3.4.2

Cancel the common factor.

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{8 (\frac{x^{7}}{8 \cdot 4782969}) + 8 (\frac{7}{8}) - 7}$

Step 5.3.4.3

Rewrite the expression.

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{\frac{x^{7}}{4782969} + 8 (\frac{7}{8}) - 7}$

Step 5.3.5

Cancel the common factor of $8$ .

Tap for more steps...

Step 5.3.5.1

Cancel the common factor.

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{\frac{x^{7}}{4782969} + 8 (\frac{7}{8}) - 7}$

Step 5.3.5.2

Rewrite the expression.

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{\frac{x^{7}}{4782969} + 7 - 7}$

Step 5.3.6

Simplify the expression.

Tap for more steps...

Step 5.3.6.1

Subtract $7$ from $7$ .

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{\frac{x^{7}}{4782969} + 0}$

Step 5.3.6.2

Add $\frac{x^{7}}{4782969}$ and $0$ .

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{\frac{x^{7}}{4782969}}$

Step 5.3.6.3

Rewrite $4782969$ as $9^{7}$ .

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{\frac{x^{7}}{9^{7}}}$

Step 5.3.7

Rewrite $\frac{x^{7}}{9^{7}}$ as ${(\frac{x}{9})}^{7}$ .

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 \sqrt[7]{{(\frac{x}{9})}^{7}}$

Step 5.3.8

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

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 (\frac{x}{9})$

Step 5.3.9

Cancel the common factor of $9$ .

Tap for more steps...

Step 5.3.9.1

Cancel the common factor.

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = 9 (\frac{x}{9})$

Step 5.3.9.2

Rewrite the expression.

$f (\frac{x^{7}}{38263752} + \frac{7}{8}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{7}}{38263752} + \frac{7}{8}$ is the inverse of $f (x) = 9 \sqrt[7]{8 x - 7}$ .

$f^{- 1} (x) = \frac{x^{7}}{38263752} + \frac{7}{8}$