Question

Let $D$ be domain of $f(x) = \log_{\left[\frac{x+1}{2}\right]}{(x^2-5x+6)}$ where, $[.]$ denotes GIF, then which of the intervals are NOT contained in $D$:

• A. $\left(\frac{1}{2},1\right]$
• B. $\left[\frac{3}{2},2\right]$
• C. $(2,3)$
• D. $\left[\frac{5}{2},\frac{7}{2}\right]$

Answer 1

Answer: (A), (B), (C), (D)

All the given intervals are NOT contained in $D$.

Answer 2

The domain of $f(x) = \log_{\left[\frac{x+1}{2}\right]}{(x^2-5x+6)}$ requires the argument $x^2-5x+6 > 0$, which gives $x \in (-\infty, 2) \cup (3, \infty)$. The base $\left[\frac{x+1}{2}\right]$ must be positive and not equal to 1. $\left[\frac{x+1}{2}\right] > 0 \implies \frac{x+1}{2} \ge 1 \implies x \ge 1$. $\left[\frac{x+1}{2}\right] \neq 1 \implies \frac{x+1}{2} < 1$ or $\frac{x+1}{2} \ge 2 \implies x < 1$ or $x \ge 3$. Combining the base conditions gives $x \in [3, \infty)$. The domain $D$ is the intersection of the argument and base conditions: $((-\infty, 2) \cup (3, \infty)) \cap [3, \infty) = (3, \infty)$. We check if each given interval is a subset of $(3, \infty)$.

• $(\frac{1}{2},1] = (0.5, 1]$ is not a subset of $(3, \infty)$.
• $[\frac{3}{2},2] = [1.5, 2]$ is not a subset of $(3, \infty)$.
• $(2,3)$ is not a subset of $(3, \infty)$.
• $[\frac{5}{2},\frac{7}{2}] = [2.5, 3.5]$ is not a subset of $(3, \infty)$.

All the given intervals are NOT contained in the domain $D$.