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Functions

A real function, or simply a function \(f:\mathcal{D}\subset\mathbb{R}\to \mathbb{R}\) returns a real number on being fed a real number. Sometimes it can be defined by a formula, e.g., as

\(\begin{align} y&=f(x), \end{align}\) where \(x\in \mathcal{D}\subset \mathbb{R}\) and \(y\) belong to \(\mathbb{R}\). A distinction is made between the function \(f\) and the function’s value \(f(x)\), or \(y\). Some examples of functions are,

\[\begin{align} y&=\sin(x),\quad x\in \mathbb{R},\\ y&=1/x\quad x\in \mathbb{R}-\{0\},\\ \end{align}\]

and so on.

Mappings

Instead of the simple operation which returns to us a real number on being fed a real number, we can think of more general operations. In those more general operations, a general mathematical object is returned to us by the operation when it is fed a general mathematical object. Examples, of general mathematical objects can be complex numbers, plane geometric objects (trinagle, circles, etc.), 3D geometric objects (cubes, spheres, ect.), digital images, and of course the main focus of this course vectors. The more generalized functions are called “mappings’’. So, functions are special cases of mappings. That is, mappings between subsets of the real number line are called functions. The only requirements on a mapping is that it should be well defined. That is, we should clearly know the domain of the mapping, and for each element of the domain the mapping should always return a well defined, unique object.

Examples

• \(A:\mathcal{D}\to \mathbb{R}^+\), where \(\mathcal{D}\) is set of all traingles, and for each \(x\in \mathcal{D}\), \(A(x)\) returns to us the area of the triangle.
• \(V:\mathcal{D}\to \mathbb{R}^+\), where \(\mathcal{D}\) is set of all cubes, and for each \(x\in \mathcal{D}\), \(V(x)\) returns to us the volume of the cube.
• \(D:\mathcal{D}\to \{\text{Mon}, \text{Tu}, \text{Wed}, \text{Th}, \text{Fri}, \text{Sat}, \text{Sun}\}\), where \(\mathcal{D}\) is set of all dates, and for each \(x\in \mathcal{D}\), \(D(x)\) returns to us which day of the week a particular date is.

• \(\text{Col}:\mathcal{D}\to \{\text{Red}, \text{Blue}, \text{Green}\}\), where \(\mathcal{D}\) is set of all pixels on a well defined digital image, and for each \(x\in \mathcal{D}\), \(\text{Col}(x)\) returns to the pixel’s color.

• \(\text{Norm}:\mathbb{R}^3\to \mathbb{R}_{\ge 0}\), where \(\mathbb{R}^3\) is the collection of all ordered sets of three real numbers, of the form $x_{\bullet}=(x_1,x_2,x_3)$. And for each \(x\in \mathcal{D}\), \(\text{Norm}(x_{\bullet})\) returns to the non-negative number $\sf{sqrt}(x_ix_i)$.

Non Examples

• \(\text{Moisture}:\mathcal{D}\to \{\text{Dry}, \text{Wet}\}\), where \(\mathcal{D}\) is set of all traingles, and for each \(x\in \mathcal{D}\), \(\text{Moisture}(x)\) returns to us whether the triangle is “dry” or “wet”.

• \(V:\mathcal{D}\to \mathbb{R}^+\), where \(\mathcal{D}\) is set of all words in english, and for each \(x\in \mathcal{D}\), \(V(x)\) returns to us the volume of the word. (concept of volume is ill defined for words)

• \(\text{Age}:\mathcal{D}\to \mathbb{R}^+\), where \(\mathcal{D}\) is set of all people on earth, and for each \(x\in \mathcal{D}\), \(\text{Age}(x)\) returns to us the age of the person’s child. (A person may have more than one child, “unique” is lost, may have no child, “always” is lost)