Question

The domain of the function $f\left(x\right)={\log }_{[x+\frac{1}{2}]}|x^2-5x+6|$ is

• A. $[\frac{3}{2},2)\cup (2,3)\cup (3,\infty )$
• B. $[\frac{3}{2},\infty )$
• C. $[\frac{1}{2},\infty )$
• D. $Noneofthese$

Answer 1

The correct option is C $[\frac{1}{2},\infty )$

$f\left(x\right)={\log }_{[x+\frac{1}{2}]}|x^2-5x+6|$

$=>(x+\frac{1}{2})>1$

Which gives $=>x>\frac{1}{2}$ $\left(1\right)$

For $|x^2-5x+6|$:

By putting it to zero the roots will be:

$x^2-3x-2x+6=0$

Gives: $(x-2)(x-3)=0$

so the $roots$ are $x=2,3$

The $Domain$ for $|x^2-5x+6|$ is all real values $\left(2\right)$

From Equation $\left(1\right)$ and $\left(2\right)$:

$x\in [\frac{1}{2},\infty )$