Q. For which value of b is the function f(x) = { 2x + 1, x < 1; b, x = 1; x^2 + 1, x > 1 continuous at x = 1?
Solution
Setting the left limit (2(1) + 1 = 3) equal to the right limit (1^2 + 1 = 2), we find b = 3.
Correct Answer: B — 2
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Q. For which value of b is the function f(x) = { x^2 - 1, x < 1; b, x = 1; 3x - 2, x > 1 continuous at x = 1?
Solution
Setting limit as x approaches 1 gives b = 2 for continuity.
Correct Answer: C — 2
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Q. For which value of b is the function f(x) = { x^2 - 4, x < 2; bx + 2, x >= 2 } continuous at x = 2?
Solution
Setting the two pieces equal at x = 2 gives us 0 = 2b + 2. Solving for b gives b = -1.
Correct Answer: B — 4
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Q. For which value of b is the function f(x) = { x^3 - 3x + b, x < 1; 2x + 1, x >= 1 continuous at x = 1?
Solution
Setting 1 - 3 + b = 2 gives b = 4 for continuity.
Correct Answer: A — 0
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Q. For which value of c is the function f(x) = { 3x + c, x < 1; 2x^2, x >= 1 continuous at x = 1?
Solution
Setting 3(1) + c = 2(1)^2 gives c = -1 for continuity.
Correct Answer: B — 0
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Q. For which value of c is the function f(x) = { x^2 - 4, x < c; 3x - 5, x >= c } continuous at x = c?
Solution
Setting the two pieces equal at x = c: c^2 - 4 = 3c - 5. Solving gives c = 3.
Correct Answer: C — 3
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Q. For which value of c is the function f(x) = { x^2 - c, x < 1; 2x + 1, x >= 1 continuous at x = 1?
Solution
Setting x^2 - c = 2x + 1 at x = 1 gives c = 2.
Correct Answer: C — 2
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Q. For which value of c is the function f(x) = { x^2, x < c; 2x + 1, x >= c continuous at x = c?
Solution
Setting x^2 = 2x + 1 at x = c gives c = 2.
Correct Answer: C — 2
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Q. For which value of m is the function f(x) = { 3x + m, x < 1; 2, x = 1; mx + 1, x > 1 continuous at x = 1?
Solution
Setting 3 + m = 2 and 2 = m + 1 gives m = 1 for continuity.
Correct Answer: B — 0
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Q. For which value of p is the function f(x) = { x^2 + p, x < 0; 3x - 1, x >= 0 } continuous at x = 0?
Solution
Setting the two pieces equal at x = 0 gives us p = -1.
Correct Answer: B — 0
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Q. If f(x) = 1/(x-1), what is the point of discontinuity?
-
A.
x = 0
-
B.
x = 1
-
C.
x = -1
-
D.
x = 2
Solution
The function is discontinuous at x = 1 because it leads to division by zero.
Correct Answer: B — x = 1
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Q. If f(x) = x^2 - 4, what is the limit of f(x) as x approaches 2?
-
A.
0
-
B.
2
-
C.
4
-
D.
Undefined
Solution
The limit as x approaches 2 is f(2) = 2^2 - 4 = 0.
Correct Answer: C — 4
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Q. If f(x) = x^3 - 3x + 2, then f(x) is continuous at:
-
A.
All x
-
B.
x = 0
-
C.
x = 1
-
D.
x = -1
Solution
f(x) is a polynomial function and is continuous for all x.
Correct Answer: A — All x
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Q. If f(x) = { 2x + 3, x < 0; kx + 1, x >= 0 } is continuous at x = 0, what is the value of k?
-
A.
-3/2
-
B.
1/2
-
C.
3/2
-
D.
2
Solution
Setting the two pieces equal at x = 0: 3 = k(0) + 1. Solving gives k = -3/2.
Correct Answer: A — -3/2
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Q. If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 } is continuous at x = 0, what is k?
Solution
For continuity at x = 0, we need the left limit (1) to equal k. Thus, k = 1.
Correct Answer: A — 1
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Q. If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x + 1, x > 0 }, what value of k makes f continuous at x = 0?
Solution
To be continuous at x = 0, k must equal the limit from the left, which is 1.
Correct Answer: B — 1
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Q. If f(x) = { x^2 + 1, x < 0; k, x = 0; 2x, x > 0 }, for f(x) to be continuous at x = 0, k must be:
Solution
For continuity at x = 0, k must equal the limit as x approaches 0, which is 1.
Correct Answer: B — 1
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Q. If f(x) = { x^2 + 1, x < 0; kx + 2, x = 0; 3 - x, x > 0 is continuous at x = 0, find k.
Solution
For continuity at x = 0, we need 1 = 2, thus k must be 1.
Correct Answer: B — 2
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Q. If f(x) = { x^2 + 1, x < 0; kx + 3, x = 0; 2x - 1, x > 0 is continuous at x = 0, find k.
Solution
For continuity at x = 0, we need 1 = 3 and 1 = -1 + 3k, solving gives k = 1.
Correct Answer: C — 1
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Q. If f(x) = { x^2, x < 0; 2x + 3, x >= 0 }, find f(0).
-
A.
0
-
B.
3
-
C.
1
-
D.
undefined
Solution
At x = 0, we use the second piece: f(0) = 2(0) + 3 = 3.
Correct Answer: B — 3
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Q. If f(x) = { x^2, x < 0; kx + 1, x = 0; 2x + 3, x > 0 is continuous at x = 0, find k.
Solution
To ensure continuity at x = 0, we set k(0) + 1 = 0^2, leading to k = 2.
Correct Answer: C — 1
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Q. If f(x) = { x^2, x < 1; kx + 1, x >= 1 } is continuous at x = 1, find k.
Solution
Setting the two pieces equal at x = 1 gives 1 = k + 1, hence k = 0.
Correct Answer: B — 1
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Q. If f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2, what is the value of f(2)?
Solution
For continuity at x = 2, f(2) must equal the limit from both sides, which is 4.
Correct Answer: B — 4
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Q. If f(x) = { x^2, x < 3; k, x = 3; 2x, x > 3 } is continuous at x = 3, what is the value of k?
Solution
For continuity at x = 3, we need limit as x approaches 3 from left (9) to equal f(3) = k, thus k = 9.
Correct Answer: C — 6
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Q. If f(x) = { x^2, x < 3; k, x = 3; 3x - 2, x > 3 } is continuous at x = 3, what is k?
Solution
For continuity at x = 3, we need k to equal the limit from both sides, which is 9.
Correct Answer: C — 8
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Q. If f(x) is continuous on [a, b], which of the following must be true?
-
A.
f(a) = f(b)
-
B.
f(x) takes every value between f(a) and f(b)
-
C.
f(x) is increasing
-
D.
f(x) is decreasing
Solution
By the Intermediate Value Theorem, a continuous function on a closed interval takes every value between f(a) and f(b).
Correct Answer: B — f(x) takes every value between f(a) and f(b)
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Q. Is the function f(x) = { e^x, x < 0; ln(x + 1), x >= 0 } continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
D.
Only right continuous
Solution
Both limits as x approaches 0 from the left and right are equal to 1, hence f(x) is continuous at x = 0.
Correct Answer: A — Yes
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Q. Is the function f(x) = { sin(x), x < 0; x^2, x >= 0 } continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Depends on x
-
D.
Not defined
Solution
Both limits as x approaches 0 from the left and right are equal to 0, hence f(x) is continuous at x = 0.
Correct Answer: A — Yes
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Q. Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
D.
Only right continuous
Solution
Both limits as x approaches 1 from the left and right are equal to 2, hence f(x) is continuous at x = 1.
Correct Answer: A — Yes
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Q. Is the function f(x) = |x|/x continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Depends on direction
-
D.
None of the above
Solution
The left limit is -1 and the right limit is 1, which are not equal. Therefore, f(x) is not continuous at x = 0.
Correct Answer: B — No
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