Question

Obtain the largest possible domain, and corresponding range, of the function f, defind by f(x)=x-2/3-x

Answer 1

Given function is $f(x)=\frac{x-2}{3-x}$ .

Now given function has numerator and denominator both continuous and denominator is $3-x$.

So given function is not defined when $3-x=0 \implies x=3$.

So, posible domain where function is defined should not contains x=3.

Hence Largest possible domain is D = $(-\infin , 3) \cup (3,\infin).$

Now when $x\in (-\infin,3) \implies x<3 \implies 3-x>0$ and when $x\in (3,\infin) \implies 3-x<0.$

Also, for $x\in (3,\infin), x-2\in (1,\infin)$ and so $f(x)\in (-\infin,-1).$

And for $x\in (-\infin, 3), f(x)=-1+\frac{1}{3-x} \in -1+(0,\infin) \in (-1,\infin).$

So, Range of given function is $(-\infin,-1)\cup(-1,\infin).$