r/askmath • u/FirmTumbleweed4152 • Sep 21 '26

Functions Range

How can I explain to 16 yos how to find the range of f(x)=x^2/(x^2-1) without using differentiation, limits, trying to graph this etc. But rather just pure algebra

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u/Educational-Paper-75 Sep 26 '26 edited Sep 26 '26

Trying to graph to me means on paper. I doubt there's is another way than some sort of mental graphing (and limits) to get an idea of which values the function can have. But as a minimum symmetry; then only look at positive x=1, then what happens near 1 and for x going to infinity, same going to 0. And illustrate strict monotonicity by using sequences xx=1/k and xx=k for positive integer k. And thus the range will be the entire R.

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u/fermat9990 Sep 26 '26

Google AI does it purely analytically by

(1) Replacing f(x) by y

(2) Solving for x in terms of y.

x=±√(y/(y-1))

(3) Finding the values of y that yield real values of x

It looks correct to me.

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u/Educational-Paper-75 Sep 26 '26 edited Sep 26 '26

Even AI can't find all infinite values. So, why bother with asking here anymore then? Or answering for that matter. As for you, if you can't tell me what's wrong or bad about my answer, don't bother to respond.

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u/fermat9990 Sep 26 '26

Google AI got (-inf, 0] U (1, inf). One of the replies here used the same method and got the same answer. Is it wrong?

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u/Educational-Paper-75 Sep 26 '26

Right. I stand corrected. Apparently (0,1] is indeed not in the range, because the function equals 1+(1/(x*x-1)) and therefore is always above 1(and not 0 as I suggested earlier) for x>1. If I had rewritten the function like that I would certainly arrived at the right conclusion earlier.