Calculus Examples

$f (x) = x^{\frac{2}{3}}$

Step 1

Write $f (x) = x^{\frac{2}{3}}$ as an equation.

$y = x^{\frac{2}{3}}$

Step 2

Interchange the variables.

$x = y^{\frac{2}{3}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y^{\frac{2}{3}} = x$ .

$y^{\frac{2}{3}} = x$

Step 3.2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{2}{3}})}^{\frac{3}{2}} = \pm x^{\frac{3}{2}}$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify ${(y^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 3.3.1.1

Multiply the exponents in ${(y^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 3.3.1.1.1

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

$y^{\frac{2}{3} \cdot \frac{3}{2}} = \pm x^{\frac{3}{2}}$

Step 3.3.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$y^{\frac{2}{3} \cdot \frac{3}{2}} = \pm x^{\frac{3}{2}}$

Step 3.3.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{3} \cdot 3} = \pm x^{\frac{3}{2}}$

Step 3.3.1.1.3

Cancel the common factor of $3$ .

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Step 3.3.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = \pm x^{\frac{3}{2}}$

Step 3.3.1.1.3.2

Rewrite the expression.

$y^{1} = \pm x^{\frac{3}{2}}$

Step 3.3.1.2

Simplify.

$y = \pm x^{\frac{3}{2}}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y = x^{\frac{3}{2}}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - x^{\frac{3}{2}}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = x^{\frac{3}{2}}$

$y = - x^{\frac{3}{2}}$

$y = x^{\frac{3}{2}}$

$y = - x^{\frac{3}{2}}$

$y = x^{\frac{3}{2}}$

$y = - x^{\frac{3}{2}}$

Step 4

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

$f^{- 1} (x) = x^{\frac{3}{2}}, - x^{\frac{3}{2}}$

Step 5

Verify if $f^{- 1} (x) = x^{\frac{3}{2}}, - x^{\frac{3}{2}}$ is the inverse of $f (x) = x^{\frac{2}{3}}$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = x^{\frac{2}{3}}$ and $f^{- 1} (x) = x^{\frac{3}{2}}, - x^{\frac{3}{2}}$ and compare them.

Step 5.2

Find the range of $f (x) = x^{\frac{2}{3}}$ .

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Step 5.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, \infty)$

Step 5.3

Find the domain of $x^{\frac{3}{2}}$ .

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Step 5.3.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt{x^{3}}$

Step 5.3.2

Set the radicand in $\sqrt{x^{3}}$ greater than or equal to $0$ to find where the expression is defined.

$x^{3} \geq 0$

Step 5.3.3

Solve for $x$ .

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Step 5.3.3.1

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

$\sqrt[3]{x^{3}} \geq \sqrt[3]{0}$

Step 5.3.3.2

Simplify the equation.

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Step 5.3.3.2.1

Simplify the left side.

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Step 5.3.3.2.1.1

Pull terms out from under the radical.

$x \geq \sqrt[3]{0}$

Step 5.3.3.2.2

Simplify the right side.

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Step 5.3.3.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 5.3.3.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x \geq \sqrt[3]{0^{3}}$

Step 5.3.3.2.2.1.2

Pull terms out from under the radical.

$x \geq 0$

Step 5.3.4

The domain is all values of $x$ that make the expression defined.

$[0, \infty)$

Step 5.4

Find the domain of $f (x) = x^{\frac{2}{3}}$ .

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Step 5.4.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt[3]{x^{2}}$

Step 5.4.2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Step 5.5

Since the domain of $f^{- 1} (x) = x^{\frac{3}{2}}, - x^{\frac{3}{2}}$ is the range of $f (x) = x^{\frac{2}{3}}$ and the range of $f^{- 1} (x) = x^{\frac{3}{2}}, - x^{\frac{3}{2}}$ is the domain of $f (x) = x^{\frac{2}{3}}$ , then $f^{- 1} (x) = x^{\frac{3}{2}}, - x^{\frac{3}{2}}$ is the inverse of $f (x) = x^{\frac{2}{3}}$ .

$f^{- 1} (x) = x^{\frac{3}{2}}, - x^{\frac{3}{2}}$

Step 6