Given the function $f:[0,1]→[0,1]$; $f(x)=x^2$, check which one(s) of the properties it has.

Question

This homework is past due, but I am still fiddling trying to figure this out.

question: I do not understand what the heck the notation of $f:[0,1] \to [0,1]$; means. I thought I did, but my repetition states otherwise.

Here is the image of the problem for better understanding

Don't even give me the answer to the question itself. Just please explain what f:[0,1] -> [0,1]; means. I thought it was all points $0 \le x \le1$. If it is, I have no idea what the heck the answer is.

Any feedback or direction is appreciated.

Answer 1

It means the domain and codomain of the function are the closed interval from $0$ to $1$. Given any $x$ such that $0 \le x \le 1$, you must have $0 \le f(x) \le 1$. This is true for $f(x)=x^2$. You don't have to have that every point in $[0,1]$ is the image of some point in the domain. If that is true, the function is surjective.

Answer 2

I thought it was all points $0 \le x \le1$. If it is, I have no idea what the heck the answer is.

The notation $[0,1]$ means all the points $0 \le x \le 1.$

The notation $f : [0,1] \to [0,1]$ is different. It means $f$ is a function that takes its input values from $[0,1]$, and its output values are (also) part of $[0,1]$.

More generally, and as someone pretty much already said in the comments, $f : A \to B$ means $f$ takes its input values from the set $A$, and all of the output values are in the set $B.$