Question

Assertion(A): $f\left(x\right)=\log (x-2)+\log (x-3)$ and $g\left(x\right)=\log (x-2)(x-3)$ then $f\left(x\right)=g\left(x\right)$
Reason (R): Two functions $f\left(x\right)$ and $g\left(x\right)$ are said to be equal if they are defined on the same domain $A$ and the co-domain $B$ as $f\left(x\right)=g\left(x\right)\forall x\in A$

• A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.
• B. Both $A$ and $R$ are true and $R$ is NOT the correct explanation of $A$.
• C. $A$ is correct and $R$ is incorrect.
• D. $A$ is incorrect and $R$ is correct.

Answer 1

The correct option is D $A$ is incorrect and $R$ is correct.

Reason is correct.
Assertion:
For f to be defined $x-2>0\text{and}x-3>0\Rightarrow x>3$
so the domain of f is $(3,\infty )$.
Now for g to be defined $(x-2)(x-3)>0\Rightarrow x\in (-\infty ,2)\cup (3,\infty )$
Thus domain of g is $(-\infty ,2)\cup (3,\infty )$.
Clearly domain of both the functions are not same, Hence $f\ne g$