Range of a rational function (feat. Constants)

[shell idol]

I was messing around on desmos a little and found this.
I've solved it myself, I just need a double-check on my work, espesially the end bit.

[shell idol]

Photo of my work

[thick matrix]

I think it's correct.

Another way to do it is to separate the fraction, which gives $f(x)=\frac x \alpha + \frac\alpha x$

Then, by letting $t = \frac x \alpha$, this is equal to $t + \frac 1 t$

Therefore, the range of f is the range of $g: t \mapsto t + \frac{1}{t}$, which you can study a bit easily than f (especially because you don't have to worry about the sign of $\alpha$).
(but at the end of the day, I'm not sure it's is quicker)

[sleek lance]

rotaroced_

[shell idol]

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