Author: haroonkhan

A function maps an input value x into an output value y = f(x). There are cases in which we want to go the other way around; i.e., given y, we would like to find a value x such that y = f(x). Actually, this is what we do whenever we want to solve an equation. For instance, given a function that evaluates the NPV of…

So far, we have considered linear, polynomial, rational, and exponential functions. From our high school math, we might recall something about trigonometric functions; since we will not use them in the following, we leave them aside. A natural way to build quite complicated, but hopefully useful, functions is function composition. Given functions g and h, we may build the…

Before we proceed in our treatment of functions, we should pause a little and discuss a fundamental feature of functions: continuity, or lack thereof. Compare the graphs of polynomial functions in Fig. 2.8 against the graph of the rational function in Fig. 2.9. There is a striking qualitative difference between the two figures; in the first case, if…

Polynomial functions involve powers like xk, where the exponent k is an integer number. We recall some fundamental rules that are quite handy when dealing with powers and should be familiar from high school mathematics: In a monomial function f(x) = αxk, the basis x is the independent variable and the exponent k is a fixed parameter. In exponential functions we reverse their roles and…

If P(x) and Q(x) are polynomial functions, the function is a rational function. In other words, a rational function is just a ratio of two polynomials. Unlike linear and polynomial functions, the domain of a rational function need not be the whole real line. We are in trouble when the denominator polynomial is zero, i.e., when Q(x) = 0. Loosely…

The next step is to consider powers of the independent variable x. A term of the form axm is called a monomial of degree m. Summing monomials, we get a polynomial function: Here n is the degree of the polynomial. A few polynomial functions are shown in Fig. 2.8. A quick glance at the three plots suggests a few observations: We define concepts like local or global minimum…

A linear affine function has the following general form: Figure 2.7 shows a few linear functions. Strictly speaking, only the first function is linear. A function f is linear if the following condition holds: Fig. 2.7 Graphs of linear (affine) functions. for arbitrary numbers αi and xi, i = 1, 2. However, this holds only when the coefficient q in (2.4) is zero. To see…

Functions are rules that map input values to output values in a well determined way. They come in many guises, depending on what is mapped on what. Generally, a function is specified as where D is the domain of the function, i.e., the set of possible input values on which the function is defined, and I is the image or range of the function, i.e.,…

Many practical problems involve permutations and combinations of objects. A first question is: Given a collection of n objects, in how many ways can we permute them? For instance, let us consider the set {a, b, c}. Since the set is quite small, we can enumerate all of the possible permutations systematically. First we consider permutations beginning…

Consider an expression like We will meet similar expressions quite often in the book, and a nice shorthand notation for this expression is which should be read as the sum of “x subscript i,” for i ranging from 1 to 4. Sometimes, the sum limits can be symbolic, as in We may even consider an infinite sum like In…