Answer to Question #114631 in Calculus for ANJU JAYACHANDRAN

Question #114631
Obtain the largest possible domain, and corresponding range, of the function f, defind by f(x)=x-2/3-x
1
Expert's answer
2020-05-18T09:26:23-0400

Given function is f(x)=x23xf(x)=\frac{x-2}{3-x} .

Now given function has numerator and denominator both continuous and denominator is 3x3-x.

So given function is not defined when 3x=0    x=33-x=0 \implies x=3.

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

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


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

Also, for x(3,),x2(1,)x\in (3,\infin), x-2\in (1,\infin) and so f(x)(,1).f(x)\in (-\infin,-1).

And for x(,3),f(x)=1+13x1+(0,)(1,).x\in (-\infin, 3), f(x)=-1+\frac{1}{3-x} \in -1+(0,\infin) \in (-1,\infin).

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


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