Which of the following functions represents the following graph?

Question

graph begins in the second quadrant and decreases until negative 2, negative 1 and increases into the first quadrant as x increases.

Choose one answer.
A. f(x) = the square root of the sum of x and 2 - 1\

B. f(x) = (x + 2)2 - 1

C. f(x) = negative 1 over x plus 2


D. f(x) = (x + 2)3 - 1

Answer 1

If by "negative 2, negative 1" you mean (-2, -1), then
(a) this point must be on the curve and
(b) the curve must be 2nd order to have both negative and positive slopes. So:

looking at (a):
(A) f(x) = f(-2) = sqrt ((-2)+ (2-1)) = sqrt(-2 +1) = sqrt (-1) = i <> -1 → (A) is not the answer

(B) f(x) = f(-2) ((-2) + 2)^2 - 1 = (0)^2 - 1 = 0 - 1 = -1 =-1 → (B) could be the answer

(C) f(x) = f(-2) = -1/((-2) + 2) = -1/0 = infinity <> -1 → (C) is not the answer

(D) f(x) = f(-2) = ((-2) + 2)^3 - 1 = 0^3 - 1 = 0 - 1 = -1 = -1 → (D) could be the answer.

looking at (b):
(B) is second order which means it will change slope once (think a parabola which is second oder → (B) could be the answer

(D) is third order, which means it will change slope once → (B) is not the answer.

So the answer is (B)