Functions

Introduction:

Let D and S to be sets. A function from D to S is a rule (formula, correspondance) that assigns an element of D a unique element of S. The set D is called domain of the function (denoted dom(f), and the set of elements of S to which elements of D are assigned is called the range of the function (denoted range(f)). See figure, and check domain and range of the given example.

A function may be given by a formula as the following example:

Since f depends on s, we say that f is a function of s. Here, dom(f) = D and range(f) = {1,2,3,4,5,6}.

Conventions:   From now on functions refers to Real valued functions. If the domain of a function is not given explicitly, the convention is to take the largest set of real numbers x such that f(x) is a real number.
 

Examples of real-valued functions:

Graph of a Function:

Let f be a function with domain D, then the graph of f is the set of all points P(x,f(x)) in the plane, where . See the graph of Example 1 above.

Example of a piecewise function:

Exercises:

1. Use Sketch to figure out the graph of a function fby plotting a few points P(x,f(x)), where x is in the dom(f). Link the points to see how the graph might look like. Try:
   • f(x) = abs(x) (ie; |x|), dom(f) = [-10,10).
   • f(x) = sqrt(x^2), dom(f) = (-8,10].
   • f(x) = 1/x; What is the dom(f)?
   
    

    

2. Now try  applet to graph some functions.
   • f(x) = abs(x) (ie; |x|), dom(f) = [-10,10).
   • f(x) = sin(x);
   • f(x) = 1/x;
   • f(x) = 1/x; 1/(1-x^2);
   • f(x) = ln(x); exp(x);
   
    
3. State the formula for the function f and give the domain of the function.
    

   f(x) is the perimeter of a circle of radius x.   f(x) is the area of a circle of radius x.    f(x) is the perimeter of a square of side x.   f(x) is the volume of a cube of side x.   f(x) is the total surface of a cube of side x.   f(x) is the hypotenuse of the right triangle whose legs have lengths 3 and x.   f(x) is the length of the side AB in the triangle in the figure below.

   Solution?? Click (1)  (2)  (3)  (4)  (5)  (6)  (7) to see solution below.   Erase solutions!!