Could someone please explain to me how you can tell if a function is one-to-one? I don't understand...
do you know what it means to be one-to-one? what is you're definition of one-to-one functions.
graphically: we can show that a graph is a one-to-one relation by the horizontal line test. drawing any horizontal line will cut the graph ONLY once
analytically: we can show a graph is one-to-one by showing that if then
or equivalently, if we can find any two none equal elements in the domain, then we would find that
example:
the function y = x + 1 is one-to-one, since if we have:
(which is )
we can subtract 1 from both sides to get:
thus the function is one-to-one
counter-example:
the function is not one-to-one
since if we have it is not necessarily true that
look:
square rooting both sides we obtain:
in general
a specific example would be, since (you could just state this example by the way to prove it's not one-to-one, no explanation needed)
so there are two elements in the domain that map to a single element in the range. thus we have a many-to-one relation, not one-to-one
that's why i said "in general."
let's say i did do that:
how many possible combination of signs can we have? well 4 of course
1.
2.
3.
4.
Of course, we realize that 1 and 3 are saying the same thing, and 2 and 4 are saying the same thing. so i can summarize these 4 possibilities into two possibilities
1. .............(this is equation 1 above, which is the same as equation 3)
2. ...........(this is equation 2 above, which is the same as equation 4)
Now I can summarize these two equations into one:
namely,
but like i said, this kind of reasoning is not necessary, saying "since , we have that is NOT one-to-one" will suffice