On-line Math 21

1.1  Modeling

You need to be familiar with a number of general classes of functions, functions which occur as models of various natural phenomena

Linear functions

These are functions of the sort f(x) = ax+b . Their graphs are lines.

Polynomials

These are important primarily because we can understand their properties quite easily, and because any function can be pretty-well approximated by a polynomial for a limited range of values of x (more about that later). A polynomial function of degree n is a function of the sort f(x) = a_nxⁿ+a_n-1x^n-1+¼+a₁x+a₀ . For example, f(x) = 3x²-2x+4 is a polynomial of degree 2.

Powers

Functions of the sort f(x) = x^a , where a can be any number. The big thing to worry about here is to keep straight the rules for exponents, such as

x1/2 = Öx,

and

x-3/2 = 1 ( Öx) 3 .

Trigonometric functions

Trigonometric functions come up in many different situations. Any model with a geometric ``piece'' will often have something to do with trigonometric functions. In particular, anything to do with a circle, or triangles, or repeated ``cycles'' of motion will use trigonometric functions somewhere. You should know the basic properties of functions like sin(x) , cos(x) , tan(x) , etc., where x is an angle in radians.

Exponential functions

These are functions of the sort f(x) = a^x . Note that the x is in the exponent. The number a is the base. We'll introduce these later, but again we expect you to be familiar with their basic properties.

The primary reason to study exponential functions is that they model many natural phenomena, that is, the behavior of many systems are expressed in terms of exponential functions. Such things as compound interest, radioactive decay, and population growth are governed by exponential functions.

One curious number always mentioned in this context is e . e is just a number, like p, which has a specific value. That letter, e , is almost-always reserved for that particular number (again, like p » 3.14159 . e is approximately 2.71828 . But e, like p, comes from a specific relationship. If you look at the graphs of the various exponential functions y = a^x for a number of choices of a , they all pass through (1,0) , since a⁰ = 1 for any positive number a . e is that choice of base so that the slope of the curve as it passes through (1,0) is 1.

Logarithms

These are the inverses of exponential functions, written f(x) = log_ax . Again, a is the base.

The text mentions ``transcendental functions'' as if they were a specific class of functions. But they are not. A function is algebraic if it is a sum or quotient of polynomials and roots (fractional powers). If a function is not algebraic, it is transcendental.

1.2  Combining functions together

You should be familiar with how to add functions together, f+g(x) = f(x)+g(x) , as well as multiplication and division of functions. The biggest concern will be composition of functions, one function followed by another. That is f(g(x)) takes x , does g to it, and then does f to the result. You will use this most often to recognize that a function, such as

sin(   ____ Ö3x+2   ),

is a composition of simpler functions, and so can be analyzed by breaking it down into those simpler components.

Copyright (c) by David L. Johnson, last modified
On 25 Apr 2000, 16:48..