AIME Polynomials and Functions 2

Question 1

For f(x) = x^2 - 4x + 7, what is the minimum value?

Accepted Answer

3

Answer

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?

Accepted Answer

0

Answer

x^4=-4 has no real solutions since x^4≥0 for all real x.

Question 3

Simplify (x^2 - 9)/(x^2 - x - 6) for x ≠ 3 and x ≠ -2.

Accepted Answer

(x+3)/(x+2)

Answer

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:

Accepted Answer

x = 2

Answer

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 ∘ h)(3).

Accepted Answer

16

Answer

(g∘h)(3)=g(h(3))=g(4)=16.

AIME - American Invitational Mathematics Examination Practice Test