A function f defined on a subset E of real numbers has a maximum M at a point a in E if M = f(a) and if, for all x in E, f(x) is less than or equal to f(a).
Therefore, M is the maximum of the set of images of f.
Example
Consider the function defined by f(x) = –x2 + 4, and represented by the parabola below:
If x=0, then f(x)=4.
For any other value of x, f(x)<4.
Therefore, the maximum of the image of the function is 4.
We can also say that 4 is the maximum of the function f.
The function defined by f(x)=x2+4, does not have a maximum, but its minimum is 4.