investigate injective, surjective, bijective function?
Hey, I have the following function: (x,y) ∈ R²
y – 2x² + 4x + 1 = 0
I first change this to y
y = 2x² – 4x -1
I should now investigate whether the function is bijective.
The script says that it is unique but not one-to-one (unfortunately it doesn't say whether it is surjective or injective).
So for me the function is neither surjective nor injective.
This is the graphic function in Geogebra:
Is there a mistake in the script? y(values) < -3 aren't hit at all. So, not all real values are hit. Thus, it's not surjective (nor injective, as one can easily see with a parabola).
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Please introduce the complete task here without interpretations on your part. I guess you haven’t fully understood the task.
Addendum: What exactly does the sample solution look like?
By the way, “unique” always means “injective”, “unique” is “bijective”.
In the context of the task, “unique” means that a y is clearly assigned to each x. And that’s the case. The function is therefore unique in y.
Okay, thanks. I thought unambiguous can mean both injectively and surjectively, the main thing one of both is fulfilled (at both then one-indicated or bijective).
So here also surjectivity is meant “unique”? The fact that it is not injective is seen directly as it is a quadratic function in which a y-value -> is assigned two x-values.
No, a surjective function must not be clear. Also note the difference to e and f, both of which are not unique assignments from x to y.
In the case of assignments, it is NOT the same as bijective. Unique means:
Each element from B has at most one partner in A and vice versa. However, it does not have to occur any element of A and B on each side.
We’re not talking about functions yet.
Thanks for clearing.
Right. This is NOT about functions, but relations. That is, it is also not about the question whether something is surjective, injective and bijective. First of all, it is clear that for each x there is exactly one y in the relation. This is, for example, no longer fulfilled at d), because there are also 2 (for example for x=0), the relation is therefore not clear and therefore it cannot be written as a function. At e) we don’t even have a y for each x (for x = 0 we don’t find anything like that).
Therefore, the definition of the term “unique” and “unambiguous” should first be carefully considered in relation to assignments. It is strictly speaking neither unambiguously = surjective NOCH unambiguous = bijective. As this is in the present script, you have to read it first.
First of all, you should make it clear that this is only about assignments, so Relations. This is the more abstract concept. The terms “surjective” and “injective” are usually used for functions (in relation, the correspondences are “right total” and “left-alone”). Each function is a relation, but there are also relationships that are not functions.
Here, however, the task is not at all asked for functions or for surjective and bijective, so it is also no wonder that this is not in the answer! You obviously assumed that an assignment is the same as a function, but this is not the case.
An assignment is clear if at most one y value is assigned to each x value – this is the case here, because for each x value there is even exactly one y value, namely y=2×2-4×-1.
In the next task (d), e.g. B. I have a relationship that
|x-y| = 1
is defined. This is not unambiguous, then both pairs (0,1) and (0,-1) fulfill this relation, because |0-1| = |0+1 = =1. ZWEI y values are thus assigned to the x value 0 . So I cannot define a function with this relation.
One significant means that it is assigned at most one y value and at most one x value for each x value. This does not mean that the relation is also a function and it does not mean that something is injective or bijective here.
That is why it is so important to keep these terms apart. If you use your existing knowledge of functions here, you get away from the way. What you are supposed to do here is much simpler: you draw quantities in the plane and formally check the conditions for the two properties “unique” and “unique”. The definitions for these properties are somewhere in the script. All that you bring about, for example, pre-knowledge from the school, you leave it outside.
There may have been confusion of values with target quantity. The exact task would actually help.
And in the exact task, you can see that this is not about functions, but that the questioner has already gone out of the way. That’s why you are sooo right – the exact task often helps in general.