Question

Which statement about the end behavior of the logarithmic function f(x) = log(x + 3) – 2 is true?

As x decreases, y moves toward the vertical asymptote at x = -3.
As x decreases, y moves toward the vertical asymptote at x = -1.
As x increases, y moves toward negative infinity.
As x decreases, y moves toward positive infinity.

Answer 1

plato boy say the answer is; A

Answer 2

(A): As x decreases, y moves towards the vertical asymptote at x = -3

Explanation: For a logarithmic function, we know that at x = 0, log(x) is undefined. As such, there is a vertical asymptote at x = 0. When we do logarithmic transformations, the vertical asymptote won't change much.

Let's use log(x + 3) - 2 as an example.

Start with log(x).
As x decreases, the y-value can never pass through the vertical asymptote, because it would be undefined at and less than the value at the vertical asymptote. So it would keep converging closer and closer towards the asymptote, but never reach there.

Now, let's transform the insides: log(x + 3).
Here, we change the properties of our domain.
Since x > 0, for log(x),
we can say: x + 3 > 0 and x > -3
Thus, our new asymptote is at x = -3.

Now, the range transformation won't change our domain. So based on this information, we can immediately cross off (C) and (D).

(B) is irrelevant, and our only viable option has to be (A).