One-to-One function?

Question

In text it states that a one to one function is “A function f is one to one if for any two range values f(u) and f(v), f(u)=f(v) implies that u=v. What exactly does this mean? I thought if there are two equal y values it is NOT a one to one function?

Answer 1

Your intuition is probably something like: if you put in two different inputs into the function, the two outputs cannot be the same.

The definition is just saying the same thing in a different way: if you put two inputs $u$ and $v$ into the function and the two outputs $f(u)$ and $f(v)$ are the same, then the two inputs that you started with must have actually been the same, that is, $u=v$.

Answer 2

Image of two different elements cannot be same. Consider $f:\mathbb R \to \mathbb R, f(x)=x^2$. Here $f(u)=1$ for what values of $u$?

Answer 3

An easy way to visualize a one-to-one function is through the "horizontal line test". Imagine a horizontal line going across the graph of a function. With a one-to-one function, the horizontal line should either a) never cross the function; or b) cross only once. This holds true regardless of where the line is positioned; you can imagine it scanning from the top to the bottom of a graph.

As for your definition, your intuition is correct but I think your confusing f(u) with u. For a one-to-one function, each value in the range corresponds to at most one value in the domain. Put another way, the function transforms some input into some output. Each input is associated with a unique output and vice-versa. Let's say f(u) equals some y in the range of the function. Let's also say the f(v) also equals y. Since y is associated with a single unique input to the function, u and v cannot be different.

Answer 4

It's just a fancy way of saying that if the function is one-to-one then each unique input gives a unique output. If two output values are the same, then the input values must be the same, otherwise it is not a one-to-one function.

A common function that isn't one-to-one is $f(x) = x^2$. In this function, when $f(x)= 4$ we have two different possibilities for $x$, $x=2$ and $x=-2$.

In elementary algebra, checking if a function is one-to-one is introduced as the horizontal line test. With this we just check and see if it is possible to draw a horizontal line through the graph of the function that intersects the function at more than one point. If it does then we can say that it is not one-to-one.

Answer 5

One to one means: For every output there is exactly one input.

Or in other words, If two times you got the same out put, then you must have had the same input.

Or in other words. If you got $f(u) = f(v)$ that means $u = v$.

=====

Or... one to one means. If $u \ne v$ then $f(u) \ne f(v)$.... so if you ever DO find yourself with $f(u) = f(v)$ the only way that can possibly happen is if $u = v$.

Basically the two statements:

i) If $u \ne v \implies f(u) \ne f(v)$ and

ii) If $f(u) = f(v) \implies u = v$

are both equivalent. ii) is just the contrapositive of i).

"Cheap food is not good" is the contrapositive of "Good food is not cheap". THey both mean exactly the same thing.

======

To do an example.

Suppose $f(x) = x^3$ we'll take it for granted that we know $f$ is one-to-one.

Suppose $f(u) = f(v) = 8$. Then what is $u$ and what is $v$?

Well $v^3 = 8$ so $v =\sqrt[3]8 = 2$. And $u^3 = 8$ so $u = \sqrt[3]8 = 2$ and .... what do you know! $u = v$! What were the odds!