Calculus Examples

$x^{- \frac{1}{3}}$

Step 1

Interchange the variables.

$x = y^{- \frac{1}{3}}$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $y^{- \frac{1}{3}} = x$ .

$y^{- \frac{1}{3}} = x$

Step 2.2

Raise each side of the equation to the power of $- 3$ to eliminate the fractional exponent on the left side.

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

Step 2.3

Simplify the exponent.

Tap for more steps...

Step 2.3.1

Simplify the left side.

Tap for more steps...

Step 2.3.1.1

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

Tap for more steps...

Step 2.3.1.1.1

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

Tap for more steps...

Step 2.3.1.1.1.1

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

$y^{- \frac{1}{3} \cdot - 3} = x^{- 3}$

Step 2.3.1.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.3.1.1.1.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$y^{\frac{- 1}{3} \cdot - 3} = x^{- 3}$

Step 2.3.1.1.1.2.2

Factor $3$ out of $- 3$ .

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

Step 2.3.1.1.1.2.3

Cancel the common factor.

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

Step 2.3.1.1.1.2.4

Rewrite the expression.

$y^{- 1 \cdot - 1} = x^{- 3}$

Step 2.3.1.1.1.3

Multiply $- 1$ by $- 1$ .

$y^{1} = x^{- 3}$

Step 2.3.1.1.2

Simplify.

$y = x^{- 3}$

Step 2.3.2

Simplify the right side.

Tap for more steps...

Step 2.3.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = \frac{1}{x^{3}}$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = \frac{1}{x^{3}}$ is the inverse of $f (x) = x^{- \frac{1}{3}}$ .

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Simplify the denominator.

Tap for more steps...

Step 4.2.3.1

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

Tap for more steps...

Step 4.2.3.1.1

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

$f^{- 1} (x^{- \frac{1}{3}}) = \frac{1}{x^{- \frac{1}{3} \cdot 3}}$

Step 4.2.3.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.3.1.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$f^{- 1} (x^{- \frac{1}{3}}) = \frac{1}{x^{\frac{- 1}{3} \cdot 3}}$

Step 4.2.3.1.2.2

Cancel the common factor.

$f^{- 1} (x^{- \frac{1}{3}}) = \frac{1}{x^{\frac{- 1}{3} \cdot 3}}$

Step 4.2.3.1.2.3

Rewrite the expression.

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

Step 4.2.3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.5

Multiply $x$ by $1$ .

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

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

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

Step 4.3.3

Change the sign of the exponent by rewriting the base as its reciprocal.

$f (\frac{1}{x^{3}}) = {(x^{3})}^{\frac{1}{3}}$

Step 4.3.4

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

Tap for more steps...

Step 4.3.4.1

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

$f (\frac{1}{x^{3}}) = x^{3 (\frac{1}{3})}$

Step 4.3.4.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.4.2.1

Cancel the common factor.

$f (\frac{1}{x^{3}}) = x^{3 (\frac{1}{3})}$

Step 4.3.4.2.2

Rewrite the expression.

$f (\frac{1}{x^{3}}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{1}{x^{3}}$ is the inverse of $f (x) = x^{- \frac{1}{3}}$ .

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