Calculus Examples

$y = {(5 x - 6)}^{\frac{1}{3}}$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Convert expressions with fractional exponents to radicals.

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Step 1.1.1

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

$\sqrt[3]{{(5 x - 6)}^{1}}$

Step 1.1.2

Anything raised to $1$ is the base itself.

$\sqrt[3]{5 x - 6}$

$\sqrt[3]{5 x - 6}$

Step 1.2

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.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

Since the domain is all real numbers, ${(5 x - 6)}^{\frac{1}{3}}$ is continuous over all real numbers.

Continuous

Step 3

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$