Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?

f(x)=(2x−3)^2/3

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The function $ f(x) = (2x - 3)^{2/3} $ is a composition of a power function and a linear function. The power function $ g(x) = x^{2/3} $ is continuous for all $ x \geq 0 $.

The inner function $ 2x - 3 $ is a linear function, which is continuous everywhere. Therefore, the domain of $ 2x - 3 $ is all real numbers.

Since $ g(x) = x^{2/3} $ is continuous for $ x \geq 0 $, the composition $ f(x) = (2x - 3)^{2/3} $ is continuous where $ 2x - 3 \geq 0 $.

Solve $ 2x - 3 \geq 0 $ to find the interval of continuity. This gives $ x \geq \frac{3}{2} $.

Since $ f(x) $ is defined for $ x \geq \frac{3}{2} $, check continuity from the right at $ x = \frac{3}{2} $. The function is continuous from the right at this endpoint.>

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. Understanding continuity is essential for determining where a function behaves predictably without breaks or jumps.

Endpoints of Intervals

Endpoints of intervals are the boundary points that define the start and end of an interval on the real number line. When analyzing continuity, it is crucial to evaluate the behavior of the function at these endpoints, as a function may be continuous from the left or right at these points, indicating one-sided limits exist and match the function's value.

Piecewise Functions and Roots

The function f(x) = (2x - 3)^(2/3) is a piecewise function that involves roots and powers. Understanding how to handle fractional exponents is important, as they can introduce points of discontinuity. Specifically, the behavior of the function near the root (where the expression inside the parentheses equals zero) must be analyzed to determine continuity and the nature of the function at that point.

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