AMC 12 Algebra & Functions 2

Question 1

If f(x) = 2x + 3 and g(x) = x^2 - 1, what is f(g(2))?

Accepted Answer

9

Answer

g(2) = 3, so f(3) = 2(3) + 3 = 9.

Question 2

What is the sum of the roots of 3x^2 - 12x + 5 = 0?

Accepted Answer

4

Answer

By Vieta's formulas the sum of roots is -b/a = 12/3 = 4.

Question 3

Solve log_2(x) + log_2(x - 2) = 3.

Accepted Answer

4

Answer

x(x - 2) = 8 gives x = 4 or x = -2, and only x = 4 keeps both logs defined.

Question 4

If f(x + 1) = x^2 + 2x for all x, what is f(3)?

Accepted Answer

8

Answer

f(x + 1) = (x + 1)^2 - 1, so f(t) = t^2 - 1 and f(3) = 8.

Question 5

What is the product of the roots of x^3 - 6x^2 + 11x - 6 = 0?

Accepted Answer

6

Answer

For a monic cubic x^3 + bx^2 + cx + d, the product of roots is -d = 6.

AMC 12 - - American Mathematics Contest 12 Practice Test