Range of a function Find the range of  

$$f(x)=\frac{(x-a)(x-b)}{(c-a)(c-b)}+\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}$$ where $a, b, c$ are distinct real numbers such $a\neq b\neq c\neq a$.


$$f(x)=\frac{(x-a)(x-b)}{(c-a)(c-b)}+\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}$$
$$f(a)= \dfrac{(a-b)(a-c)}{(a-b)(a-c)}=1$$
$$f(b)= \frac{(b-c)(b-a)}{(b-c)(b-a)}=1$$
$$f(c)= \frac{(c-a)(c-b)}{(c-a)(c-b)}=1$$
Now what is $f'(x)?$, $f'(x)=\dfrac{2x-(a+b)}{(c-a)(c-b)}+\dfrac{2x-(b+c)}{(a-b)(a-c)}+\dfrac{2x-(c+a)}{(b-c)(b-a)}$. Now watch the house of cards fall. Then you have a constant function, and you are aware of that constant. 
