Can this be an onto function? 
A function $f:R\to R$, where $R$ is set of real numbers, is defined by $$f(x)=\frac{\alpha x^2 + 6x - 8}{\alpha + 6x -8x^2}$$
Find the interval of values of $\alpha$ for which $f$ is onto. Justify your answer.

My attempt:
Since the domain is all real values, the denominator in $f(x)$ must have no real roots, so discriminant of the quadratic (denominator) is less than zero :
$$9 + 8\alpha <0 {\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}..(1)$$
But then I noticed that the expression for discriminant is the same for both numerator and denominator, if $(1)$ is true, numerator will also have no real roots which means there is no $x∈R$ such that $f(x)=0$ $\\\\\\$ and so $f(x)$ can never be onto for any value of $\alpha$
Textbook Solution:
The expression can be written as (Note: $f(x)=y$):
$$ (8y+\alpha)x^2 + 6(1-y)x-(8+\alpha y)=0{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}..(1)$$
Since $x$ is real, $D\ge0$:
$$(8\alpha +9)y^2 + (46 + {\alpha}^2)y + (8\alpha + 9)\ge 0{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}{\space}..(2)$$
For $y$ to be real, $9+8\alpha\gt0$ and $(46+{\alpha}^2)^2-4(9+8\alpha)^2\le0$
Solving these inequalities, we get $2\le\alpha\le14$
I've looked for solutions online but they all give the textbook answer which led me to believe that maybe I was doing something wrong. Please help me.
 A: The question is poorly worded. They do not want the domain to be the set of real numbers, they want it to be $\{x \in \mathbb{R} : \alpha + 6x - 8x^2 \neq 0\}$.
In calculus and analysis it's common for functions not to be defined everywhere and many times the phrase "real valued function" means the domain is an appropriate subset of the real numbers. The proper term for this is a "partial function" but since these are so common in this area of mathematics, it's common to just call these "functions."
Whether it's still appropriate to write "$f : \mathbb{R} \to \mathbb{R}$" I would argue no. I've always seen $f : D \subseteq \mathbb{R} \to \mathbb{R}$ where $D$ is an unspecified set but is assumed to be the largest subset of $\mathbb{R}$ for which the definition of $f$ makes sense. It would seem that whoever wrote this textbook have a different opinion than me.
I would add: while I think it's appropriate for a calculus book to be a bit sloppy about domains, if this were a book on elementary set theory/logic/basic proofs, then being this sloppy is wholly inappropriate.
