Math Test CH.2
Q1. Which of the following is true?
A. If a function 𝑓𝑓 has a vertical asymptote at 𝑥𝑥 = 𝑐𝑐, then f is not defined at that point
B. If 𝑓𝑓 is continuous at some point, it can be redefined at that point
C. A rational function can be continuous for all 𝑥𝑥 values
D. None
Q2. Evaluate 𝑙𝑙𝑙𝑙𝑙𝑙𝜋𝜋
𝑥𝑥→
2
𝜋𝜋
−𝑥𝑥
2
A. −1
.
1+𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥
.
𝑥𝑥→𝜋𝜋 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥
Q3. Evaluate 𝑙𝑙𝑙𝑙𝑙𝑙
6
3−3 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥
Q4. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 , find lim
3
A. 2
𝑥𝑥→2
A. −
1
2
B. −
3
2
𝑓𝑓(𝑥𝑥)−𝑓𝑓(2)
.
2−𝑥𝑥
B. 0
C. 1
2
3
D. −
B. 0
C.
C.
D. DNE
1
2
D. DNE
2
3
Q5. According to the intermediate value theorem (IVT), determine the function where the equation 𝑓𝑓(𝑥𝑥) = 0 has a
root between −1 and 3
A. 𝑓𝑓(𝑥𝑥) = 7𝑥𝑥 2 + 10𝑥𝑥 + 3
B. 𝑓𝑓(𝑥𝑥) = −4𝑥𝑥 2 + 11𝑥𝑥 + 3
2
𝑥𝑥
D. None
C. 𝑓𝑓(𝑥𝑥) = + 1
Q6. According to the intermediate value theorem (IVT), determine whether or not the equation 𝑓𝑓(𝑥𝑥) = 0 has a root
between [0 , 4], where 𝑓𝑓 is shown on the right.
A. 𝑓𝑓(𝑥𝑥) = 0 has a root according to IVT
B. IVT fails due to 𝑓𝑓(0) and 𝑓𝑓(4) both being positive
C. IVT fails due to a discontinuity on [0 , 4]
D. IVT fails due to 𝑓𝑓(0) and 𝑓𝑓(4) both being negative
Q7. The figure on the right shows a rectangle and an isosceles triangle inscribed in a circle of diameter 10.
What is the value of 𝑥𝑥 that will make the areas of the rectangle and the triangle equal?
A. 2
1
B. 2
C. 4
D. 1
Q8. The function 𝑑𝑑(𝑡𝑡) = −16𝑡𝑡 2 + 144 represents a model to determine the position of an object falling from a
height of 144 meters, 𝑡𝑡 seconds after the fall. The limit 𝑙𝑙𝑙𝑙𝑙𝑙
𝑡𝑡→𝑎𝑎
𝑓𝑓(𝑡𝑡)−𝑓𝑓(𝑎𝑎)
represents the velocity of the falling object at
𝑡𝑡−𝑎𝑎
𝑡𝑡 = 𝑎𝑎, what will the velocity of the object be when it hits the ground?
A. 96𝑚𝑚/𝑠𝑠 going down
B. 96𝑚𝑚/𝑠𝑠 going up
2𝑥𝑥 − 1
3
Q9. Find the discontinuity interval of the function 𝑓𝑓(𝑥𝑥) = �𝑥𝑥
A. no discontinuity
B. [−1 , 0]
𝑥𝑥 2 − 1
𝑥𝑥
Q10. Find the discontinuity interval of the function 𝑓𝑓(𝑥𝑥) = �1
A. {0}
B. [2 , 3[
𝑥𝑥
C. 0𝑚𝑚/𝑠𝑠
𝑥𝑥 < −1
|𝑥𝑥 − 1| ≤ 1 .
𝑥𝑥 > 3
D. cannot be determined
C. ]2 , 3]
D. [−1 , 0] ∪ ]2 , 3]
C. {0} ∪ [2 , 3[
D. no discontinuity
− 1 < 𝑥𝑥 < 2
|𝑥𝑥 − 1| ≥ 2
Q11. Which of the following functions has a removable discontinuity at 𝑥𝑥 = 0 and a non-removable discontinuity at
𝑥𝑥 = 2 and is continuous for all other values of 𝑥𝑥 ?
A. 𝑓𝑓(𝑥𝑥) =
C. 𝑓𝑓(𝑥𝑥) =
𝑥𝑥∗|𝑥𝑥−2|
𝑥𝑥−2
𝑥𝑥
B. 𝑓𝑓(𝑥𝑥) = (𝑥𝑥−2)∗𝑠𝑠𝑠𝑠𝑠𝑠
𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥
𝑥𝑥 2 −2𝑥𝑥
D. 𝑓𝑓(𝑥𝑥) =
𝑥𝑥
𝑥𝑥 3 −2𝑥𝑥 2
𝑥𝑥
Q12. Which of the following functions has a removable discontinuity at 𝑥𝑥 = −2 and a non-removable discontinuity
at 𝑥𝑥 = 1 and vanishes at 𝑥𝑥 = 0 ?
A. 𝑓𝑓(𝑥𝑥) =
C. 𝑓𝑓(𝑥𝑥) =
�𝑥𝑥 2 +2𝑥𝑥�(𝑥𝑥−1)
𝑠𝑠𝑠𝑠𝑠𝑠(𝑥𝑥+2)
B. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 +𝑥𝑥−2
𝑥𝑥 2 +3𝑥𝑥+2
3−𝑐𝑐𝑐𝑐𝑐𝑐(𝑥𝑥+2)
𝑥𝑥 2 +𝑥𝑥−2
Q13. Redefine the function 𝑓𝑓(𝑥𝑥) =
𝑥𝑥−1
+1
𝑡𝑡𝑡𝑡𝑡𝑡(𝑥𝑥−1)
A. 𝑓𝑓(𝑥𝑥) = �
D. 𝑓𝑓(𝑥𝑥) =
𝑥𝑥 2 +𝑥𝑥−2
𝑥𝑥−1
+ 1 to be continuous at 𝑥𝑥 = 1 .
𝑡𝑡𝑡𝑡𝑡𝑡(𝑥𝑥−1)
𝑥𝑥−1
+1
𝑡𝑡𝑡𝑡𝑡𝑡(𝑥𝑥−1)
B. 𝑓𝑓(𝑥𝑥) = �
𝑥𝑥 ≠ 1
2
𝑥𝑥 = 1
0
𝑥𝑥 = 1
𝑥𝑥−1
+1
C. 𝑓𝑓(𝑥𝑥) = �𝑡𝑡𝑡𝑡𝑡𝑡(𝑥𝑥−1)
�𝑥𝑥 2 +2𝑥𝑥�|𝑥𝑥−1|
1
𝑥𝑥 ≠ 1
D. cannot be redefined
𝑥𝑥 ≠ 1
𝑥𝑥 = 1
Q14. Which of the following is false?
A. 𝑙𝑙𝑙𝑙𝑙𝑙
𝑥𝑥→0
|𝑥𝑥−5|−|𝑥𝑥+5|
𝑥𝑥
= −2
𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥−𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥
.
𝑐𝑐𝑐𝑐𝑐𝑐(2𝑥𝑥)
𝑥𝑥→
Q15. Evaluate 𝑙𝑙𝑙𝑙𝑙𝑙𝜋𝜋
4
√4+𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥−2
.
𝑥𝑥
𝑥𝑥→0
Q16. Evaluate 𝑙𝑙𝑙𝑙𝑙𝑙
1−𝑐𝑐𝑐𝑐𝑐𝑐(2𝑥𝑥)
=2
𝑥𝑥 2
𝑥𝑥→0
B. 𝑙𝑙𝑖𝑖𝑖𝑖
√2
A. 2
A. 0
𝑥𝑥
B. 1
Q17. Evaluate 𝑙𝑙𝑙𝑙𝑙𝑙 𝑐𝑐𝑐𝑐𝑐𝑐2 𝑥𝑥−𝑠𝑠𝑠𝑠𝑛𝑛2 𝑥𝑥−𝑐𝑐𝑐𝑐𝑐𝑐(2𝑥𝑥) . A. 0
𝑥𝑥→0
Q18. If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 + 𝑎𝑎 and
A. (±2 , 0)
D. None
C. √2
C.
B. 2
D. DNE
1
4
D.
C. 1
1
2
B. �± , 0�
A. ]−1 , 2]
B. [−1 , 2]
A. −3
B. 2
𝑥𝑥 + 1
Q20. Given the function 𝑓𝑓(𝑥𝑥) = � 2
𝑥𝑥 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐
Karo-33
𝑥𝑥−2
.
𝑥𝑥+1
1
2
D. DNE
1−√1+𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥
𝑥𝑥
≤ 𝑓𝑓(𝑥𝑥) ≤ |𝑥𝑥−2|−|𝑥𝑥+2| as 𝑥𝑥 approaches 0, find the x-intercept(s) of 𝑓𝑓 .
𝑥𝑥
Q19. Find the discontinuity interval of the function 𝑓𝑓(𝑥𝑥) = �
Grade12_ENG
√2
B. − 2
√1+𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥
= 𝐷𝐷𝐷𝐷𝐷𝐷
𝑥𝑥→𝜋𝜋 𝜋𝜋−𝑥𝑥
C. lim
√2
C. �±√2 , 0�
D. �± 2 , 0�
C. [−1 , 2[
D. ]−1 , 2[
C. 1
D. 4
1 < 𝑥𝑥 < 3
is continuous, find 𝑓𝑓(𝑏𝑏 + 𝑐𝑐) .
|𝑥𝑥 − 2| ≥ 1