For f(x) given below, find the numbers at which f is discontinuous. At which of these values is f continuous from the right, from the left, or neither? f(x)={2x if x≤1 3−x if 14

Question

For f(x) given below, find the numbers at which f is discontinuous. At which of these values is f continuous from the right, from the left, or neither?

f(x)={2x	if x≤1
3−x	if 1<x≤4

√
x
	if x>4

BlackTom AI · Accepted Answer

We start by rewriting the piecewise function clearly to see where the formula changes:
f(x) = { 2x, for x ≤ 1; 3 − x, for 1 < x ≤ 4; √x, for x > 4 }.

First, identify potential breakpoints where the formula changes. Those are at x = 1 and x = 4 because the definitions shift there.

Examine x = 1:
- Left-hand value as x → 1− comes from the first piece, so f(x) = 2x. Thus the left-hand limit is 2·1 = 2, and f(1) itself equals 2 (since x ≤ 1 uses 2x, so f(1) = 2).
- Right-hand value as x → 1+ comes from the second piece, so f(x) = 3 − x. Thus the right-hand limit is 3 − 1 = 2.
- Since the left-hand limit, the right-hand limit, and f(1) all agree at 2, there is no discontinuity at x = 1; the function is continuous there from both sides.

Examine x = 4:
- Left-hand value as x → 4− comes from the second piece, so f(x) = 3 − x. The left-hand limit is 3 − 4 = −1. The function value at x = 4 is determined by the rule for 1 < x ≤ 4, which gives f(4) = 3 − 4 = −1.
- Right-hand value as x → 4+ comes from the third piece, so f(x) = √x. The right-hand limit is √4 = 2.
- The left-hand limit equals f(4) (−1), but the right-hand limit is 2, which is different. This means a jump in the limit occurs at x = 4, so the function is discontinuous there.

Now determine left- and right-continuity at the discontinuity x = 4:
- Left-continuity requires lim_{x→4−} f(x) = f(4). This holds since both are −1, so the function is continuous from the left at x = 4.
- Right-continuity requires lim_{x→4+} f(x) = f(4). Here the right-hand limit is 2 while f(4) is −1, so the function is not continuous from the right at x = 4.

Summary of findings (conceptual flow): the only potential breakpoints are at x = 1 and x = 4. At x = 1, all one-sided limits and the function value match, so no discontinuity and certainly no issue with one-sided continuity. At x = 4, the left side matches the function value, but the right side does not, creating a discontinuity with left-continuity but not right-continuity.

Note on the provided options: the answer choices field appears to be empty, so this reasoning covers how to determine the discontinuities and one-sided continuities for the given f(x) without relying on predefined options.

MATH-112-301-001 Unproctored Midcourse 1 Practice Exam 1

For f(x) given below, find the numbers at which f is discontinuous. At which of these values is f continuous from the right, from the left, or neither? f(x)={2x if x≤1 3−x if 1<x≤4 √ x if x>4

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