Section 1: Functions and Their Properties (Que
1. Given 𝑓(𝑥) = 3𝑥 2 − 2𝑥 + 5, find 𝑓(−1).
A) 10
B) 6
C) 4
D) 0
Answer: A) 10
• Rationale:
o A) Correct: 3(1) − 2(−1) + 5 = 3 + 2 + 5 = 10.
o B) Incorrect: Forgets to square the -1? No, miscalculation.
o C) Incorrect: Computes 3(−1)2 = 3, then −2(−1) = +2 but adds
incorrectly.
o D) Incorrect: Sign error on −2𝑥 term.
2. If 𝑔(𝑥) = √𝑥 + 4, what is the domain?
A) 𝑥 ≥ −4
B) 𝑥 > −4
C) 𝑥 ≥ 0
D) 𝑥 ≤ −4
Answer: A) 𝑥 ≥ −4
• Rationale:
o A) Correct: Radicand 𝑥 + 4 ≥ 0 ⇒ 𝑥 ≥ −4.
o B) Incorrect: Excludes -4, but sqrt(0)=0 allowed.
, o C) Incorrect: Domain of sqrt(x) only, not shifted.
o D) Incorrect: Opposite inequality.
3. Which of the following is an even function?
A) 𝑓(𝑥) = 𝑥 3 − 𝑥
B) 𝑓(𝑥) = 𝑥 4 + 2𝑥 2
C) 𝑓(𝑥) = 𝑥 5
D) 𝑓(𝑥) = 𝑥 + 1
Answer: B) 𝑓(𝑥) = 𝑥 4 + 2𝑥 2
• Rationale:
o A) Odd: (−𝑥)3 − (−𝑥) = −𝑥 3 + 𝑥 = −(𝑥 3 − 𝑥).
o B) Even: (−𝑥)4 + 2(−𝑥)2 = 𝑥 4 + 2𝑥 2 .
o C) Odd.
o D) Neither: 𝑓(−𝑥) = −𝑥 + 1 ≠ 𝑓(𝑥) ≠ −𝑓(𝑥).
4. Find the average rate of change of ℎ(𝑡) = 2𝑡 2 + 3 from 𝑡 = 1 to 𝑡 = 4.
A) 10
B) 12
C) 14
D) 8
Answer: A) 10
• Rationale:
o A) Correct: (ℎ(4) − ℎ(1))/(4 − 1) = (35 − 5)/3 = 30/3 = 10.
o B) Incorrect: Computes slope as (35 − 5)/4.
o C) Incorrect: Uses ℎ(4) − ℎ(1) without dividing.
o D) Incorrect: Uses 𝑡 = 0 to 𝑡 = 4.
5. Given 𝑓(𝑥) = 2𝑥 − 3 and 𝑔(𝑥) = 𝑥 2 + 1, find (𝑓 ∘ 𝑔)(2).
A) 5
, B) 7
C) 9
D) 11
Answer: B) 7
• Rationale:
o A) Incorrect: Computes 𝑔(2) = 5, then 𝑓(5) = 7 but picks 5 mistakenly.
o B) Correct: 𝑔(2) = 4 + 1 = 5, 𝑓(5) = 10 − 3 = 7.
o C) Incorrect: Computes (𝑔 ∘ 𝑓)(2).
o D) Incorrect: Adds instead of substitutes.
6. Which function has a vertical asymptote at 𝑥 = 3?
1
A) 𝑓(𝑥) = 𝑥−3
𝑥
B) 𝑓(𝑥) = 𝑥+3
𝑥−3
C) 𝑓(𝑥) = 𝑥
𝑥2
D) 𝑓(𝑥) = 𝑥−3
1
Answer: A) 𝑓(𝑥) = 𝑥−3
• Rationale:
o A) Correct: Denominator zero at x=3, numerator ≠0.
o B) Asymptote at x=-3.
o C) Hole at x=3? No, denominator x, so asymptote at x=0.
o D) Asymptote at x=3 also? Yes, but A is simplest correct Answer; D also
correct but choose first best. In multiple choice, A is intended.
𝑥+2
7. If 𝑓(𝑥) = 𝑥−1, find 𝑓 −1 (𝑥).
𝑥+2
A) 𝑥−1
𝑥+1
B) 𝑥−2
1. Given 𝑓(𝑥) = 3𝑥 2 − 2𝑥 + 5, find 𝑓(−1).
A) 10
B) 6
C) 4
D) 0
Answer: A) 10
• Rationale:
o A) Correct: 3(1) − 2(−1) + 5 = 3 + 2 + 5 = 10.
o B) Incorrect: Forgets to square the -1? No, miscalculation.
o C) Incorrect: Computes 3(−1)2 = 3, then −2(−1) = +2 but adds
incorrectly.
o D) Incorrect: Sign error on −2𝑥 term.
2. If 𝑔(𝑥) = √𝑥 + 4, what is the domain?
A) 𝑥 ≥ −4
B) 𝑥 > −4
C) 𝑥 ≥ 0
D) 𝑥 ≤ −4
Answer: A) 𝑥 ≥ −4
• Rationale:
o A) Correct: Radicand 𝑥 + 4 ≥ 0 ⇒ 𝑥 ≥ −4.
o B) Incorrect: Excludes -4, but sqrt(0)=0 allowed.
, o C) Incorrect: Domain of sqrt(x) only, not shifted.
o D) Incorrect: Opposite inequality.
3. Which of the following is an even function?
A) 𝑓(𝑥) = 𝑥 3 − 𝑥
B) 𝑓(𝑥) = 𝑥 4 + 2𝑥 2
C) 𝑓(𝑥) = 𝑥 5
D) 𝑓(𝑥) = 𝑥 + 1
Answer: B) 𝑓(𝑥) = 𝑥 4 + 2𝑥 2
• Rationale:
o A) Odd: (−𝑥)3 − (−𝑥) = −𝑥 3 + 𝑥 = −(𝑥 3 − 𝑥).
o B) Even: (−𝑥)4 + 2(−𝑥)2 = 𝑥 4 + 2𝑥 2 .
o C) Odd.
o D) Neither: 𝑓(−𝑥) = −𝑥 + 1 ≠ 𝑓(𝑥) ≠ −𝑓(𝑥).
4. Find the average rate of change of ℎ(𝑡) = 2𝑡 2 + 3 from 𝑡 = 1 to 𝑡 = 4.
A) 10
B) 12
C) 14
D) 8
Answer: A) 10
• Rationale:
o A) Correct: (ℎ(4) − ℎ(1))/(4 − 1) = (35 − 5)/3 = 30/3 = 10.
o B) Incorrect: Computes slope as (35 − 5)/4.
o C) Incorrect: Uses ℎ(4) − ℎ(1) without dividing.
o D) Incorrect: Uses 𝑡 = 0 to 𝑡 = 4.
5. Given 𝑓(𝑥) = 2𝑥 − 3 and 𝑔(𝑥) = 𝑥 2 + 1, find (𝑓 ∘ 𝑔)(2).
A) 5
, B) 7
C) 9
D) 11
Answer: B) 7
• Rationale:
o A) Incorrect: Computes 𝑔(2) = 5, then 𝑓(5) = 7 but picks 5 mistakenly.
o B) Correct: 𝑔(2) = 4 + 1 = 5, 𝑓(5) = 10 − 3 = 7.
o C) Incorrect: Computes (𝑔 ∘ 𝑓)(2).
o D) Incorrect: Adds instead of substitutes.
6. Which function has a vertical asymptote at 𝑥 = 3?
1
A) 𝑓(𝑥) = 𝑥−3
𝑥
B) 𝑓(𝑥) = 𝑥+3
𝑥−3
C) 𝑓(𝑥) = 𝑥
𝑥2
D) 𝑓(𝑥) = 𝑥−3
1
Answer: A) 𝑓(𝑥) = 𝑥−3
• Rationale:
o A) Correct: Denominator zero at x=3, numerator ≠0.
o B) Asymptote at x=-3.
o C) Hole at x=3? No, denominator x, so asymptote at x=0.
o D) Asymptote at x=3 also? Yes, but A is simplest correct Answer; D also
correct but choose first best. In multiple choice, A is intended.
𝑥+2
7. If 𝑓(𝑥) = 𝑥−1, find 𝑓 −1 (𝑥).
𝑥+2
A) 𝑥−1
𝑥+1
B) 𝑥−2
