Math Is Fun Forum

xXxEmZyxXx

Member

Registered: 2007-10-15

Posts: 12

functions

x/(1+x^2) is this surjective injective or both how can you tell???

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: functions

You have to state the domain and the codomain. Otherwise you can’t tell. Anyway, let’s assume that your function is from ℝ to ℝ. Sketch the curve. Now determine how it intersects horizontal lines y = a.

(i) If for every a, the line y = a intersects the curve only once (or not at all), then the function is injective.

(ii) If for every a, the line y = a intersects the curve at least once, then the function is surjective.

In the case of your function, neither of the above conditions is satisfied (the line y = 0.25 instersects it twice, while the line y = 1 does not intersect it); therefore the function is neither injective nor surjective.

However, if you restrict the domain to [−1,1] (so it’s a function from [−1,1] to ℝ), then the function would be injective. Similarly, if you restrict the codomain to [−0.5,0.5] (so it’s a function from ℝ to [−0.5,0.5]), then it would be surjective. Now, do you see why it’s important to state the domain and codomain of a function in order to determine whether it is injective or surjective?

Last edited by JaneFairfax (2007-10-30 09:22:49)