1. ## Range and Domain

$\displaystyle f(x)=\left\{\begin{array}{cc}2,&\mbox{if}x\leq -3\\2x+3, & \mbox{if}-3<x<0\\1, &\mbox{if}x\geq \end{array}\right.$

In This function How can i find the domain and Range of f(x)

should i go through the graph method or any other method.please explain how can i do this?

2. ## Re: Range and Domain

Do you know what "domain" and "range" mean?? The domain of a function, f(x), is the set of all possible values of x. Some times that is specified explicitely. For example, if I define "f(x)= x^2 for x> 0". That tells me directly that the domain is x> 0. If it is not given explicitely, if we are just given a formula, it is the set of all values of x for which we can use the formula. If f is given by "$\displaystyle f(x)= \frac{x- 3}{x+ 2}$", I see that I can calculate that for every value of x except x= -2 because if x= 2, I would be dividing by 0 which is undefined. Here, we are told that is $\displaystyle x\le -3$, f(x)= 2. Okay, that's a constant and certainly exists for all such x. I can see that is $\displaystyle -3< x< 0$, f(x)= 2x+ 3. That can be calculated for all such x. Finally, we are told that if $\displaystyle x\ge 0$ (that value is missing but I assume that is what is meant) f(x)= 1. Again a constant and "calculable" for all such x. Since the three intervals, $\displaystyle x\le -3$, $\displaystyle -3< x< 0$, and $\displaystyle 0\le x$ include all numbers, the domain is "all real numbers".

The "range" is the set of all function values- if we write y= f(x), then it is the set of all y values. Obviously "2" and "1" are values of the function. The middle part, y= 2x+ 3, is a linear function from y= f(-3)= -6+ 3= -3 to y= f(0)= 3. That is, all numbers between -3 and 3 are values of y and so in the domain. Since 2 and 1 are in that set, we don't need to mention them separately. The range is "$\displaystyle -3\le y\le 3$" or, in interval notation, [-3, 3].

3. ## Re: Range and Domain

Thank you for you explain and i understand.
Thanks....