Image of a function:

Question

Why is the image of a function written in the following way?

$$Im=\{ y \in Y\, | \, y=f(x) \quad 'for\, some' \, \, x \in X\} $$

Why not '$\forall $' ?

Answer 1

Take a function like $x^2$ from $\mathbb R$ into $\mathbb R$. Then $4$ is in the image, since for example $2^2=4$. That is to say, $\exists x\in\mathbb R,\ x^2=4$. Is it the case that $\forall x\in \mathbb R,\ x^2=4$ ?

Answer 2

OK, suppose it indeed said:

$$Im=\{ y \in Y\, | \, y=f(x) \quad 'for\, all' \, \, x \in X\} $$

Now let's consider a simple function, e.g. $f(x) = 2x$, as defined over the natural numbers. So then:

$$Im=\{ y \in N\, | \, y=f(x) \quad 'for\, all' \, \, x \in N\} $$

So, for a number like $2$ to be in $Im$, it would have to be true that $2 = f(1)$ and that $2 = f(2)$ and $3=f(3)$ and ... This is not true, so $2$ is not in the image. In fact, with your definition, the Image will be completely empty, since $y=f(n)$ will not be true for all $n$. But that is clearly not what we want. Conceptually, the Image clearly should be all the even numbers.

With the official definition it works. Since we have some number $x$ (namely, $x=1$) for which $f(x)=2$, $2$ will go in the image, as desired.