Alisa Duarte

2022-03-19

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

Donovan Acevedo

The question is poorly worded. They do not want the domain to be the set of real numbers, they want it to be $\left\{x\in \mathbb{R}:\alpha +6x-8{x}^{2}\ne 0\right\}$.
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 R for which the definition of f makes sense. It would seem that whoever wrote this textbook have a different opinion than me.

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