I have a question about onto and one-to-one.?
Determine(with an explanation) whether this function is one-to-one and whether it is onto.
f:R-->RxR, where f(x)=(x^2, 2x+1). I honestly know this is probably very simple but I have not idea where to start. I'm not even sure what this function means. Can someone please help? I am so confused.
Thank you to anyone that can help!
Of course! I'll explain what one-to-one and onto mean before we analyze the given function.
A function is considered "one-to-one" (or injective) if each input value corresponds to a unique output value. In other words, no two different inputs can produce the same output. Mathematically, if f(a) = f(b), then a = b for all elements "a" and "b" in the domain of the function.
On the other hand, a function is considered "onto" (or surjective) if every element in the codomain is related to at least one element in the domain. In simpler terms, every possible value of the function's codomain is reached by some element in the domain.
Now, let's analyze the given function f: R --> R x R, where f(x) = (x^2, 2x + 1). Here, R represents the set of real numbers.
To determine if the function is one-to-one, we need to check if f(a) = f(b) implies a = b for all values of a and b in the domain. In this case, the domain is all real numbers (R).
Let's assume f(a) = f(b). Therefore, (a^2, 2a + 1) = (b^2, 2b + 1). This implies that a^2 = b^2 and 2a + 1 = 2b + 1. Considering the second equation, we can see that it simplifies to 2a = 2b, which further reduces to a = b.
Hence, we have shown that if f(a) = f(b), then a = b. Therefore, the function is one-to-one.
Now, let's determine whether the function is onto. To do this, we need to determine if every element of the codomain, R x R, can be reached by some element in the domain, R.
In our case, the codomain is the set of all ordered pairs of real numbers. Each ordered pair is of the form (x^2, 2x + 1) for some x in R.
Considering the nature of the codomain, it is clear that for any ordered pair (a, b) in R x R, we can find an input value x = √a that satisfies f(x) = (a, b). So the function f is onto.
To summarize:
- The function f: R --> R x R, where f(x) = (x^2, 2x + 1) is one-to-one because f(a) = f(b) implies a = b.
- The function f is also onto because it can reach every element in the codomain, R x R, through appropriate input values.
I hope this explanation clarifies the concepts of one-to-one and onto for you! If you have any further questions, feel free to ask.