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(t)=(t^2−1)^3/2

Verified step by step guidance

1. The function \( f(t) = (t^2 - 1)^{3/2} \) involves a square root, which requires the expression inside the root to be non-negative. Therefore, solve \( t^2 - 1 \geq 0 \) to find the domain.
2. Solve \( t^2 - 1 \geq 0 \) by factoring it as \( (t - 1)(t + 1) \geq 0 \). Determine the intervals where this inequality holds by testing values in the intervals \((-\infty, -1)\), \([-1, 1]\), and \((1, \infty)\).
3. The function is continuous on the intervals where \( t^2 - 1 \geq 0 \). From the inequality solution, identify the intervals where the function is defined and continuous.
4. Check the endpoints \( t = -1 \) and \( t = 1 \) to determine if the function is continuous from the left or right. Evaluate the limit of \( f(t) \) as \( t \) approaches these points from the left and right.
5. Based on the analysis, conclude on which intervals the function is continuous and specify at which endpoints it is continuous from the left or right.>

Key Concepts

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 jumps or breaks.

Endpoints of Intervals

Endpoints of intervals are the boundary points that define the start and end of an interval. When analyzing continuity, it is crucial to evaluate the behavior of the function at these endpoints, as a function can be continuous from the left or right at these points. This means checking the limits from either side of the endpoint to see if they match the function's value at that point.

Piecewise Functions and Domain Restrictions

Some functions may have restrictions on their domains that affect their continuity. For example, the function f(t) = (t^2 - 1)^(3/2) is defined only when the expression inside the square root is non-negative. Identifying these restrictions is vital for determining the intervals of continuity, as they dictate where the function is valid and continuous.