Suppose fÝxÞ = Ýx?1Þ 2 x+1 if x < 2 x 2 ?2x?8 x?4 if 2 ² x < 4 1 x?3 + 5 if 4 ² x . Identify any points of discontinuity, and determine (giving reasons) if they are removable, infinite (essential), or jump discontinuities. (4 Points)
Question
Suppose fÝxÞ = Ýx?1Þ 2 x+1 if x < 2 x 2 ?2x?8 x?4 if 2 ² x < 4 1 x?3 + 5 if 4 ² x . Identify any points of discontinuity, and determine (giving reasons) if they are removable, infinite (essential), or jump discontinuities. (4 Points)
Solution
The function f(x) is defined in three parts:
- f(x) = (x-1)^2 + 1 for x < 2
- f(x) = x^2 - 2x - 8 for 2 ≤ x < 4
- f(x) = x - 3 + 5 for x ≥ 4
To find the points of discontinuity, we need to check the points where the function changes its definition, which are x = 2 and x = 4.
At x = 2:
- From the left (x < 2), the limit is (2-1)^2 + 1 = 2.
- From the right (2 ≤ x < 4), the limit is 2^2 - 2*2 - 8 = -4. Since the two limits are not equal, there is a jump discontinuity at x = 2.
At x = 4:
- From the left (2 ≤ x < 4), the limit is 4^2 - 2*4 - 8 = 4.
- From the right (x ≥ 4), the limit is 4 - 3 + 5 = 6. Again, the two limits are not equal, so there is another jump discontinuity at x = 4.
Therefore, the function f(x) has jump discontinuities at x = 2 and x = 4.
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