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*(\(x\)) = –\(x^{2}\) + 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 \(f(x) = x^{2} + 4\), does not have a maximum, but its minimum is 4.