Q. If f(x) = x^3 - 3x + 2, what is the value of f(1) and is it continuous?
-
A.
0, Continuous
-
B.
0, Not Continuous
-
C.
1, Continuous
-
D.
1, Not Continuous
Solution
f(1) = 1^3 - 3(1) + 2 = 0. Since f(x) is a polynomial, it is continuous everywhere.
Correct Answer:
A
— 0, Continuous
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Q. Is the function f(x) = 1/(x-1) continuous at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
D.
Only right continuous
Solution
The function f(x) = 1/(x-1) is not defined at x = 1, hence it is discontinuous at that point.
Correct Answer:
B
— No
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Q. Is the function f(x) = sqrt(x) continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
The function f(x) = sqrt(x) is continuous at x = 0 as it is defined and the limit exists.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = |x| continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
D.
Only right continuous
Solution
The function f(x) = |x| is continuous at x = 0 because the left limit, right limit, and f(0) all equal 0.
Correct Answer:
A
— Yes
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Q. The function f(x) = 1/(x-1) is continuous on which of the following intervals?
-
A.
(-∞, 1)
-
B.
(1, ∞)
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C.
(-∞, ∞)
-
D.
(-∞, 0)
Solution
The function f(x) = 1/(x-1) is discontinuous at x = 1, hence it is continuous on (1, ∞).
Correct Answer:
B
— (1, ∞)
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Q. The function f(x) = 2x + 1 is continuous at which of the following intervals?
-
A.
(-∞, ∞)
-
B.
(0, 1)
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C.
(1, 2)
-
D.
(2, 3)
Solution
f(x) = 2x + 1 is a linear function and is continuous over the entire real line (-∞, ∞).
Correct Answer:
A
— (-∞, ∞)
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Q. The function f(x) = 2x + 3 is continuous at which of the following intervals?
-
A.
(-∞, ∞)
-
B.
[0, 1]
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C.
[1, 2]
-
D.
[2, 3]
Solution
f(x) = 2x + 3 is a linear function and is continuous over the entire real line (-∞, ∞).
Correct Answer:
A
— (-∞, ∞)
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Q. The function f(x) = 2x + 3 is continuous at which of the following? (2023)
-
A.
x = -1
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B.
x = 0
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C.
x = 1
-
D.
All of the above
Solution
f(x) = 2x + 3 is a linear function and is continuous for all x.
Correct Answer:
D
— All of the above
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Q. The function f(x) = x^2 + 3 is continuous for which of the following intervals? (2023)
-
A.
(-∞, ∞)
-
B.
(0, 1)
-
C.
(1, 2)
-
D.
(2, 3)
Solution
f(x) = x^2 + 3 is a polynomial function and is continuous for all x in (-∞, ∞).
Correct Answer:
A
— (-∞, ∞)
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Q. The function f(x) = x^2 is continuous at which of the following points? (2023)
-
A.
x = -1
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B.
x = 0
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C.
x = 1
-
D.
All of the above
Solution
The function f(x) = x^2 is a polynomial function and is continuous at all points, including -1, 0, and 1.
Correct Answer:
D
— All of the above
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Q. The function f(x) = x^3 - 3x is continuous at which of the following points? (2023)
-
A.
x = -2
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B.
x = 0
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C.
x = 2
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D.
All of the above
Solution
The function f(x) = x^3 - 3x is a polynomial function and is continuous at all points, including -2, 0, and 2.
Correct Answer:
D
— All of the above
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Q. The function f(x) = { x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
D.
Only right continuous
Solution
Left limit as x approaches 1 is 2, right limit is 1, but f(1) = 2. Hence, it is discontinuous at x = 1.
Correct Answer:
B
— No
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Q. The function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 } is:
-
A.
Continuous
-
B.
Not continuous
-
C.
Continuous from the left
-
D.
Continuous from the right
Solution
The left limit as x approaches 0 is 0, but the right limit is 1. Hence, it is not continuous at x = 0.
Correct Answer:
B
— Not continuous
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Q. The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
D.
Only right continuous
Solution
At x = 0, lim x→0- f(x) = 0 and lim x→0+ f(x) = 1, hence it is discontinuous at x = 0.
Correct Answer:
B
— No
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Q. What can be said about the function f(x) = |x| at x = 0?
-
A.
Continuous
-
B.
Discontinuous
-
C.
Only left continuous
-
D.
Only right continuous
Solution
The function f(x) = |x| is continuous at x = 0 since both left and right limits equal f(0) = 0.
Correct Answer:
A
— Continuous
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Q. What is the continuity of the function f(x) = sqrt(x) at x = 0? (2022)
-
A.
Continuous
-
B.
Not continuous
-
C.
Only left continuous
-
D.
Only right continuous
Solution
The function f(x) = sqrt(x) is continuous at x = 0 as it is defined and the limit exists.
Correct Answer:
A
— Continuous
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Q. Which of the following functions is continuous at all points?
-
A.
f(x) = 1/x
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B.
f(x) = x^3
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C.
f(x) = sqrt(x)
-
D.
f(x) = tan(x)
Solution
f(x) = x^3 is a polynomial function, which is continuous everywhere.
Correct Answer:
B
— f(x) = x^3
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Q. Which of the following functions is continuous at x = 0?
-
A.
f(x) = 1/x
-
B.
f(x) = e^x
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C.
f(x) = tan(x)
-
D.
f(x) = 1/(x^2 + 1)
Solution
The function f(x) = e^x is continuous everywhere, including at x = 0.
Correct Answer:
B
— f(x) = e^x
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Q. Which of the following functions is continuous on the interval [0, 1]?
-
A.
f(x) = 1/x
-
B.
f(x) = x^3
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C.
f(x) = sqrt(x)
-
D.
f(x) = 1/(x-1)
Solution
f(x) = x^3 is a polynomial function and is continuous on the interval [0, 1].
Correct Answer:
B
— f(x) = x^3
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Q. Which of the following statements is true about the function f(x) = 1/(x-1)? (2022)
-
A.
Continuous at x = 1
-
B.
Continuous everywhere
-
C.
Not continuous at x = 1
-
D.
Continuous at x = 0
Solution
The function f(x) = 1/(x-1) is not continuous at x = 1 because it is undefined there.
Correct Answer:
C
— Not continuous at x = 1
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Q. Which of the following statements is true about the function f(x) = 1/(x-3)?
-
A.
Continuous at x = 3
-
B.
Continuous everywhere
-
C.
Not continuous at x = 3
-
D.
Continuous at x = 0
Solution
The function f(x) = 1/(x-3) is not defined at x = 3, hence it is not continuous at that point.
Correct Answer:
C
— Not continuous at x = 3
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Q. Which of the following statements is true about the function f(x) = |x|?
-
A.
Continuous everywhere
-
B.
Discontinuous at x = 0
-
C.
Continuous only at x = 1
-
D.
Discontinuous everywhere
Solution
The function f(x) = |x| is continuous everywhere, including at x = 0.
Correct Answer:
A
— Continuous everywhere
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Q. Which of the following statements is true regarding the function f(x) = 1/(x-3)?
-
A.
Continuous at x = 3
-
B.
Discontinuous at x = 3
-
C.
Continuous everywhere
-
D.
Discontinuous everywhere
Solution
The function f(x) = 1/(x-3) is discontinuous at x = 3 because it is undefined at that point.
Correct Answer:
B
— Discontinuous at x = 3
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Q. Which of the following statements is true regarding the function f(x) = |x|?
-
A.
Continuous everywhere
-
B.
Discontinuous at x = 0
-
C.
Continuous only for x > 0
-
D.
Discontinuous for x < 0
Solution
The function f(x) = |x| is continuous everywhere, including at x = 0.
Correct Answer:
A
— Continuous everywhere
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