



 
Hi, In looking for the domain of a function $f(x)$ it is often easier to look for values of $x$ that are not in the domain of $f(x).$ That is for functions from the real numbers to the real numbers you look for values of $x$ so that $f(x)$ does not produce a real number. Thus for example if $f(x) = \sqrt{x}$ then $x$ cannot be negative. For all other values of $x,\; f(x)$ is a real number so the domain of $f(x) = \sqrt{x}$ is all nonnegative real numbers. For your function $g(x) = x(x1)$ are there any real numbers $x$ so that $g(x)$ does not produce a real number? Again for the range of a function $f(x)$ it is often easier to look for numbers $y$ which you never get as an outcome of the function $f(x).$ Thus for example if $f(x) = x^2$ then you never get a negative value of $f(x)$ but you can get all nonnegative numbers. Hence the range of $f(x)$ is all nonnegative numbers. For the range of $g(x)$ I would prefer to write the expression $y = x(x  1).$ I am going to show you two ways to approach this, one algebraically because I think that is what is expected and one graphically because I think it is easier, especially if you have software or a graphing calculator that will graph the function. Geometric approach.
Algebraic approach.
I suggest you solve the problem using both approach to see if your answers agree. I hope this helps,  


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