Suppose,
f(x) = { (x - 1)^2 / x + 1 if x < 2
(x^2 - 2x - 8)/(x - 4) if
2 </= x < 4
(1 / (x - 3)) + 5) if 4 </=
x
Identify any points of discontinuity, and determine (giving reasons) if they are removable, infinite, or jump discontinuities.
When I graphed this function on my calculator, I found that there was definitely a jump discontinuity at x=4. There is, as far as I can see, one more discontinuity at -1, it looks as though it might be an infinite discontinuity since it is a vertical asymptote. Is this correct?
That is correct. f(-1) = 4/0 which is infinite.
To identify points of discontinuity in a function, you need to check the conditions that determine the function's behavior in different intervals.
Let's analyze the given function, f(x), interval by interval:
1. For x < 2:
In this interval, the function f(x) is defined as (x - 1)^2 / (x + 1).
There are no points of discontinuity in this interval.
2. For 2 ≤ x < 4:
In this interval, the function f(x) is defined as (x^2 - 2x - 8) / (x - 4).
There are no points of discontinuity in this interval.
3. For x ≥ 4:
In this interval, the function f(x) is defined as (1 / (x - 3)) + 5.
Here, x = 4 is a point of discontinuity since the denominator becomes zero, resulting in an undefined value.
Based on your observation, you correctly identified that x = 4 is a point of jump discontinuity. This is because the function approaches different finite values (defined values) from the left and right sides of x = 4, causing a jump in the graph.
Moreover, you mentioned a possible discontinuity at -1. To confirm if it is a point of discontinuity, we need to check all the intervals individually.
1. For x < 2:
As stated earlier, there are no points of discontinuity in this interval.
2. For 2 ≤ x < 4:
Again, there are no points of discontinuity in this interval.
3. For x ≥ 4:
In this interval, x = -1 is not within the given range, so it is irrelevant for this analysis.
Therefore, x = -1 is not a point of discontinuity in the defined domain of the function.
To summarize:
- The function f(x) has a jump discontinuity at x = 4.
- There are no other points of discontinuity within the defined domain.
You correctly identified the jump discontinuity at x = 4, and there are no infinite discontinuities in this function.