Suppose 𝑔(𝑥)={1𝑥−2𝑖𝑓 𝑥<12𝑥−3𝑖𝑓 𝑥≥1.g(x)={ x−21 2x−3 if x<1if x≥1 .The best description concerning the continuity of g(x) is that the function:A.is continuous.B.has a jump discontinuity.C.has an infinite discontinuity.D.has a removable discontinuity.E.None of the above
Question
Suppose 𝑔(𝑥)={1𝑥−2𝑖𝑓 𝑥<12𝑥−3𝑖𝑓 𝑥≥1.g(x)={ x−21 2x−3 if x<1if x≥1 .The best description concerning the continuity of g(x) is that the function:A.is continuous.B.has a jump discontinuity.C.has an infinite discontinuity.D.has a removable discontinuity.E.None of the above
Solution
The function g(x) is defined as two different functions depending on whether x is less than 1 or greater than or equal to 1.
For x < 1, g(x) = x - 2. This is a linear function and is continuous for all x.
For x >= 1, g(x) = 2x - 3. This is also a linear function and is continuous for all x.
The point of interest is at x = 1. We need to check if the function is continuous at this point.
The limit of g(x) as x approaches 1 from the left (using the first function) is 1 - 2 = -1.
The limit of g(x) as x approaches 1 from the right (using the second function) is 2*1 - 3 = -1.
Since the two one-sided limits are equal, the limit of g(x) as x approaches 1 exists and is -1.
The value of the function at x = 1 is g(1) = 2*1 - 3 = -1 (using the second function because x is equal to 1).
Since the limit of g(x) as x approaches 1 is equal to the value of the function at x = 1, the function is continuous at x = 1.
Therefore, the function g(x) is continuous for all x, so the best answer is A. The function is continuous.
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