Question

what can you say about the end behavior of the function $f(x)=\log_2(4x + 8)+16$?
a. both ends decrease
b. one end decreases and one end approaches a constant
c. both ends increase
d. one end increases and one end decreases

Explanation:

Step1: Find domain of the function

The argument of the logarithm must be positive: $4x + 8 > 0 \implies x > -2$. So domain is $(-2, +\infty)$.

Step2: Analyze end behavior

As $x \to -2^+$: $4x + 8 \to 0^+$, so $\log_2(4x + 8) \to -\infty$, thus $f(x) \to -\infty$ (decreases).
As $x \to +\infty$: $4x + 8 \to +\infty$, so $\log_2(4x + 8) \to +\infty$, thus $f(x) \to +\infty$ (increases).

Step3: Match with options

One end (left) decreases, one end (right) increases.

Answer:

D. One end increases and one end decreases