A function \(f\) defined on a subset

*E*of real numbers has a minimum*m*at a point \(a\) in E if*m*= \(f(a)\) and if, for all \(x\) in E, \(f(x)\) is greater than or equal to \(f(a)\).Therefore,

*m*is the minimum 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 minimum of the image of this function is 4.

We can also say that 4 is the minimum of the function *f*.

The function defined by \(f(x)\) = –\(x^{2}\)+ 4, does not have a minimum, but its