a color image can be defined as a function of two variables, (x, y), where x and y are the spatial coordinates of a pixel in the image. The function returns a vector of three values, corresponding to the red, green, and blue (RGB) intensities of the pixel.

Mathematically, a color image can be represented as a function f(x, y) = (r(x, y), g(x, y), b(x, y)), where r(x, y), g(x, y), and b(x, y) are the red, green, and blue intensities of the pixel at coordinates (x, y).

The range of the function f(x, y) is the set of all possible RGB triples. In practice, the RGB values are typically restricted to a finite range, such as [0, 255].

The color image function can be used to represent any color image, regardless of its size or resolution. It can also be used to perform operations on color images, such as image filtering, image compression, and image segmentation.

So a color image is also a function. It's just that the value at each location is no longer a single number that represents the light intensity. Instead, it can be a vector. A vector that holds three different intensities for three color components.

For instance, each location in an RGB has values for Red, Green, and Blue. These values can be separated into three different channels or planes. If you take all of the red intensity values in an image, you have your red plane. Similarly, if you take all the green intensity values, you get your green plane, likewise, for blue. Given this scheme where each location is associated with a tuple of three values and assuming that each of these intensity values is a real number, and so are the values for x and y. How would you represent this image as a functional mapping? Mark the expressions that correctly represent the desired functional mapping. 

Let's first look at the range of the function. Since each location has a tuple of three values, hence the right-hand side of the mapping should have R cross R cross R, or R cubed. The first expression is therefore clearly wrong. Note that a color image is still a mapping over a domain of two dimensional x and y values. Hence the third expression is also incorrect. Therefore, we can represent an RGB image as a mapping from R cross R to R cross R cross R. This is the same as writing R cross R maps to R cubed. In fact, you could also shorten it to R squared to R cubed

Soluton