- #1

brotherbobby

- 649

- 156

- Homework Statement
- Find the possible values that the following expression can take : ##\dfrac{x^2-x-6}{x-3}##

- Relevant Equations
- 1. For a (rational) function of the form ##y=\frac{f(x)}{g(x)}##, we should have the function ##g(x)\ne 0\; \forall \; x##. This would determine the domain of the function ##\mathscr{D}##

2. If we can invert the function ##y=f(x)## to obtain ##x=f^{-1}(y)##, the allowable values of ##y## would determine the range of the function ##\mathscr{R}##.

**Let me copy and paste the problem as it appears in the text :
Problem statement :**

**Attempt 1 (from text) :**The book and me independently could solve this problem. I copy and paste the solution from the book below.

**Attempt 2 (my own) :**The problem should afford a solution using the second idea I put in the Relevant Equations above - namely that the range of the function can be obtained by inverting it. So let ##y=x^2−x−6x−3⇒x^2−x−6=xy−3y⇒x^2−x(1+y)+3(y−2)=0##. For the values of x to be real ##(x∈R)##, the discriminant of the function ##D=b^2−4ac≥0⇒(1+y)^2−12(y−2)≥0⇒y^2−10y+25≥0⇒(y−5)^2≥0##. But the square of a number is always greater than 0. Hence y, which is the expression, can take all values : ##y∈R##.

**Issue :**Clearly my answer misses out on y≠5. This could have been achieved if the discriminant D>0. However, the discriminant can be equal to 0 also, in which case we would only have one real root were the quadratic expression was set to zero.

A help or a hint would be welcome.