Notation: $f(A)$ when $f$ is a function $f:A\to B$.

Question

I've seen the following notation with no previous clarification: $f(A)$, when $f$ is a function $f:A\to B$.

Am I correct to assume $f(A)$ should be the image of $f$?

E: I'd appreciate downvoters would comment their opinions about this question.

Answer 1

For $f:A \rightarrow B$

$f(A)= \{ f(x) | x \in A \}$

Answer 2

$f(A)$ is the image of the function. Sometimes also called the range of the function in high school.

It is the set of element $b\in B$ so that $b=f(a)$ for some $a\in A$.

for example, if $f(x)=x^2$ is defined with domain $\mathbb R$ and co-domain $\mathbb R$ then $f(\mathbb R)$ is $[0,\infty)$, that is, the image of $\mathbb R$ is the set of non-negative numbers, that is because the negative numbers are not the square of a real. While the non-negative real numbers are.

Answer 3

It usually means the image of A. Set-theorists use a different notation. The problem is that a member of a set A can also be a subset of A. So if $B \subset A=$ domain$(f)$ and $B \in A$ then "$f(B)$" is ambiguous.And "$fB$" for the image of $B$ is inadequate because if you want the image of $B \cap C$, then $fB \cap C$ and $f(B \cap C)$ are both ambiguous. They use $Y=f(B)$ to mean that $(B,Y)$ belongs to the graph of $f$. For the image of $B$, they write $f$"$(B)$, read "$f$ double-tick $B$". Including the case when $B$ is the domain of $f$.