Consider the Following: Thursday Sept 25, 1997
Prof. Harel Barzilai

  • The distance to your destination, at different times during a car trip.

  • The size of the ozone hole changes betweeen different seasons, years, and continents/hemispheres.

  • The pressure of a gas in a jar depends on the temperature.

  • The carrot yield of a farm depends on the population of rabbits in the area.
    ...And the population of wolves...
    ...And the population of worlf parasites...
    ...And the seasonal rainfall...
    ...And the temperature...
    ...And the mineral content of the soil...[etc]...

    These are all examples of Functions

    (I) One way to express a functional relationship (or function ) is verbally as above.

    (II) Another way to represent a function is through a series of measurements -- numerically.

    (III) One can express a function graphically by (a) A plot of the type of data in (II) (point plots) or (b) A "graph" of the familiar type, in the x-y-plane.

    (IV) A Formula like y=f(x) where "f(x)" is given as an explicit formula, is a mathematical model of representation of I, II, and IIIa (while IIIb is a graphical representation of the formula).

    Nature does not hand us IV on a silver platter! But formuals can be powerful tools for modeling and then analyzing real-world functional relationships.

    Types of Functions

  • Polynomials, like 3x7+7x2 -8. This is a polynomial of degree 7.

    A polynomial of degree...is the same as what we call...
    Zero A Constant-function
    One A Line or linear function
    Two A Quadraric
    Three A Cubic

  • The Rational functions are the functions R(x) defined as quotients P(x)/Q(x) of two polynomials.

  • Review the trigonometric, inverse-trigomestric, exponential and logarithmic functions in section 1.2.

  • The absolute-value function is an example of a piecewise-defined function, since the familiar absolute value function |x| can also be written as:
                    {  -x   if x < 0
           |x|= {   0   if x = 0
	        {   x   if x > 0

  • We say that a function f(x) is increasing on an interval I (for example,
    I=(c,d)={x| c<x<d} in set notation) if f(a)<f(b) always holds when a<b in I. Similarly we say that f(x) is decreasing on the interval I if f(a)>f(b) always holds when a>b in I.

    A function f(x) is even if f(-x) equals f(x) for all x. A function is odd if f(-x) equals -f(x) for all x. Please note that while all positive integers are either even or odd, most functions are neither even nor odd. Classify the following as even, odd, or neither:

    x2       x3       x+1       sin(x)       cos(x)       5x2-x+2       |x|

  • Review shifting, streching, reflecting of the graphs of functions, pages 32-34, on your own. Read carefully also the materials on combining two functions. E.g f+g(x), f-g(x), fg(x), and f/g(x).

  • Another way to combine two functions f(x) and g(x) is to compose the two functions. The composition f(g) or f¤g, which is defined by f¤g(x) being set to equal f(g(x)).

    For example, if pressure P is a function of temperature T, say P=f(T), and temperature T is a function of time, say T=g(t), then the composite function is h=f¤g which gives pressure P as a function h of t, where h(t)=[f¤g](t)=f(g(t)).