Question

The set of values of $x$ for which the function $f\left(x\right)={\log }_{[x+\frac{1}{2}]}|x^2-5x+6|$ is defined is
(where [.] denote the greatest integer function)

• A. $[\frac{3}{2},2)$
• B. $(2,3)$
• C. $(3,\infty )$
• D. $[2,\infty )$

Answer 1

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

The correct options are
A $[\frac{3}{2},2)$
B $(2,3)$
C $(3,\infty )$

$f\left(x\right)={\log }_{[x+\frac{1}{2}]}|x^2-5x+6|$
The function is defined only when,
$x^2-5x+6\ne 0,[x+\frac{1}{2}]>0,[x+\frac{1}{2}]\ne 1$

$x^2-5x+6\ne 0\Rightarrow (x-2)(x-3)\ne 0\Rightarrow x\ne 2,3\cdots \left(1\right)$

$[x+\frac{1}{2}]>0\Rightarrow x\ge \frac{1}{2}\cdots \left(2\right)$

$[x+\frac{1}{2}]\ne 1\Rightarrow x\notin [\frac{1}{2},\frac{3}{2})\cdots \left(2\right)$

Using equation $\left(1\right),\left(2\right)$ and $\left(3\right)$,
$x\in [\frac{3}{2},2)\cup (2,3)\cup (3,\infty )$