AIME Polynomials and Functions 2 ā Questions and Answers
Question 1: For f(x) = x^2 - 4x + 7, what is the minimum value?
- 3 (Correct answer)
- 7
- 4
- 0
Correct answer: 3
Complete the square: f(x)=(x-2)^2+3; minimum value is 3 at x=2.
Question 2: How many real roots does x^4 + 4 = 0 have?
- 0 (Correct answer)
- 2
- 4
- 1
Correct answer: 0
x^4=-4 has no real solutions since x^4ā„0 for all real x.
Question 3: Simplify (x^2 - 9)/(x^2 - x - 6) for x ā 3 and x ā -2.
- (x+3)/(x+2) (Correct answer)
- (x-3)/(x-2)
- (x+3)/(x-2)
- (x-3)/(x+2)
Correct answer: (x+3)/(x+2)
Factor numerator as (x-3)(x+3) and denominator as (x-3)(x+2); cancel (x-3) to get (x+3)/(x+2).
Question 4: The function f(x) = 3/(x-2) has a vertical asymptote at:
- x = 2 (Correct answer)
- x = 3
- x = 0
- x = -2
Correct answer: x = 2
A vertical asymptote occurs where the denominator is zero: x-2=0 gives x=2.
Question 5: If g(x) = x^2 and h(x) = x + 1, find (g ā h)(3).
- 16 (Correct answer)
- 10
- 13
- 9
Correct answer: 16
(gāh)(3)=g(h(3))=g(4)=16.
Question 6: Find the product of the roots of 2x^2 - 6x + 3 = 0.
- 3/2 (Correct answer)
- 3
- 6
- 1/2
Correct answer: 3/2
By Vieta's, product of roots = c/a = 3/2.
AIME Polynomials and Functions 2
For f(x) = x^2 - 4x + 7, what is the minimum value?