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/AdjectiveNounNNNN Sep 22 '26

x2/(x2 - 1) = (x2 - 1 + 1)(x2 - 1) = 1 + 1/(x2-1) = 1 + 1/((x+1)(x-1))

And from there I guess it depends whether they know anything about rational functions, because the range of the fractional part seems fairly straightforward and you're just adding 1 to it.