How many discontinuities are in the function f(x), where f(x) = -1 when x < -1, and f(x) = x for values of x ≥ -1?
Question
How many discontinuities are in the function f(x), where f(x) = -1 when x < -1, and f(x) = x for values of x ≥ -1?
Solution
The function f(x) is defined in two parts: f(x) = -1 for x < -1 and f(x) = x for x ≥ -1.
Step 1: Identify the points of discontinuity A function is discontinuous at a point where it is not defined or where it cannot be made continuous by filling in a single point.
Step 2: Check the point x = -1 For x < -1, f(x) = -1 and for x ≥ -1, f(x) = x. At x = -1, the left-hand limit is -1 and the right-hand limit is -1. Since the left-hand limit equals the right-hand limit, the function is continuous at x = -1.
Step 3: Check for other discontinuities Since the function is defined for all other values of x, there are no other points of discontinuity.
So, the function f(x) has no discontinuities.
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