For real x , the function f(x) =(x−a)(x−b)(x−c)  will be onto function provided

y=f(x) =(x−a)(x−b)(x−c)  is onto ⇒ x2−(y+a+b)x+(ab+cy)=0 …(1) has real roots for all y∈R ⇒ (y+a+b)2−4(ab+cy)≥0    ∀ y∈R ⇒y2+2(a+b−2c)y+(a−b)2≥0    ∀y∈R ⇒ 4(a+b−2c)2+4(a−b)2≤0 ⇒(c−a)(c−b)≤0 ⇒ c  lies between a  and b ⇒ a≤c≤b  or a≥c≥b .