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 (essential), or jump discontinuities. How do I find the continuities in a piece wise function?
aside from the obvious asymptotes,
just check where the definition changes.
does f(2) equal the left and right limits?
Just use x=2 and 4 and evaluate the different pieces.
For example, from the left, f(2) = (2-1)^2/(2+1) = 4/3
watch for holes (when f(x) = 0/0)
on the right, f(2) = (4-4-8)/(4-2) = -4
so, there's a break at x=2
To identify points of discontinuity and determine their type (removable, infinite, or jump discontinuity) in a piecewise function, you need to examine the function's definition at different intervals and look for any potential breaks or abrupt changes.
In your given piecewise function, f(x), we have three separate intervals:
1. For x < 2:
f(x) = (x-1)^2 / (x+1)
2. For 2 <= x < 4:
f(x) = (x^2 - 2x - 8) / (x - 4)
3. For x >= 4:
f(x) = 1 / (x - 3) + 5
Now, let's examine each interval to identify any points of discontinuity:
Interval 1 (x < 2):
In this interval, the function f(x) is well-defined and continuous for all values of x since the expression (x-1)^2 / (x+1) is valid and does not have any breaks or division by zero issues. There are no points of discontinuity in this interval.
Interval 2 (2 <= x < 4):
In this interval, the function f(x) has a potential point of discontinuity when x = 4. This is because the expression (x^2 - 2x - 8) / (x - 4) could result in division by zero when x = 4. To determine whether this point of discontinuity is removable, infinite, or a jump discontinuity, we need to evaluate the function around this point.
To evaluate the behavior of f(x) as x approaches 4 from both sides, consider the limits:
Limit of f(x) as x approaches 4 from the left (x → 4^-):
lim(x → 4^-) [(x^2 - 2x - 8) / (x - 4)] = 0
Limit of f(x) as x approaches 4 from the right (x → 4^+):
lim(x → 4^+) [(x^2 - 2x - 8) / (x - 4)] = undefined (division by zero)
Since the left and right limits are not equal, there is a jump discontinuity at x = 4. It means that as x approaches 4 from the left, f(x) approaches 0, but as x approaches 4 from the right, f(x) becomes undefined.
Interval 3 (x >= 4):
In this interval, the function f(x) is well-defined and continuous for all x values greater than or equal to 4 because the expression 1 / (x - 3) + 5 is valid and does not result in any breaks or division by zero issues. There are no points of discontinuity in this interval.
To summarize:
- There are no points of discontinuity in the interval x < 2.
- There is a jump discontinuity at x = 4 in the interval 2 <= x < 4.
- There are no points of discontinuity in the interval x >= 4.
To find the continuities in a piecewise function, you need to check the continuity of each piece individually and then consider any transition points between the pieces, as we did in the given example.