Chapter 2: Functions and Their Graphs

Functions and Their Graphs – Review

Let 31 and 31 be two sets different than the empty set. A correspondence assigning each element of the set 31, one and only one element of the set 31 is called a function from the set 31 into the set 31. Functions are generally denoted by the lower case letters such as 31. In this case a function defined from the set 31 to the set 31 is denoted by 31 or 31. In this context, the set 31 is called the domain of the function, and the set 31 is called the range set.

For a given function 31 and an element 31 from its domain 31, the element corresponding to 31 under the rule 31 is called the image of the element 31, and is denoted by 31.

Example: Let the function f:31 , 31 be given. Let find the numbers f(1) and f(311). If we write 1 instead of x in the rule of f then it holds,

31

If we write 31 instead of x in the rule of f then, we obtain the image of 31 under the function f as

31

For the function 31, the set of the images for every elements in 31 is called the image (range) of 31 and is denoted by 31. Symbolically we have

31

In order two functions to be equal the domains, the images, and the rules of these functions must be the same. In terms of mathematical notation, given two functions 31 and g31, if 31, 31, and for all 31, 31 then the functions 31 and 31 are said to be equal, and this equality is denoted by 31. For example the functions 31, 31 and 31, 31 are not equal since their domains are not same although the ranges the rules are the same.

If a rule defining a function is given but the domain has not been specified explicitly, then the largest set which makes the rule meaningful is understood. This set is denoted by 31 and is called the natural domain of the function. For instance, the function given by the rule 31 makes sense if its denominator is not zero, i.e. 31 should be satisfied. Thus, the domain 31 is should be taken as 31.

Consider the function 31. For every 31 if 31 implies 31

then the function 31 is called one-to-one (or injective). Equivalently, to confirm the one-to-oneness of a function it will be enough to show that 31 whenever 31. That is, if the images of two different elements from the domain under the action of 31 are also different, such functions are called one-to-one.

If the image is equal to the range, i.e. 31 the function 31 is called surjective (onto).

If a function which is both one-to-one and onto is called a bijection. As an example, consider 31.

Types of the Function

• For every element 31 from the domain 31, and 31, if 31, the function 31 is called constant.

• The function from A to A assigning every element of 31 to itself is called the identity function. The identity function is generally denoted by 31 or if the domain needs to be specified by 31. In this case, we write

31

• Functions, which are represented by different formulas on different subsets of its domain are called piecewise defined functions. As a piecewise defined function, absolute value is represented as 31 31, 31

  Example: 31

31

is called Dirichlet function.

Composites of functions: Let the functions 31 and 31 be given. The function 31, defined by the rule

31

is called the composition of the functions 31 and 31.

Example: Consider the functions 31, 31 and 31, 31. Let us now find the composition 31 :

For 31, using the definition we obtain

31

Inverse function: Let the bijective function 31 be given. The inverse of f is defined as

31

Here, x is the one and only element that satisfies the equality f(x)=y.

Operations on real valued functions: Given the functions 31 and 31, the addition, subtraction and multiplication of these functions are defined as

31 31 31

A little more care has to be taken defining the division. Since it is not allowed to divide by zero, the function in the denominator must not be zero for the definition to make sense. Therefore, if for every element 31, 31 then the division of f and g is defined as

31

We may define, also, multiplication by a constant. Assuming 31, we define

31

Graphs of Functions

We now introduce the Cartesian coordinate system, a tool which enables us to plot the graphs of functions.

We call the set of ordered pairs of real numbers the product of 31 by itself. We denote it by

31

The first element x of the ordered pair (x, y) is called the first component of the ordered pair, and the second element y is called the second component.

Two real lines intersecting perpendicularly at both their zeros constitutes the Cartesian coordinate system. In this system, the horizontal real line is called the x-axis, or abscissa, and the vertical real line is called the y-axis, or ordinate. We call the point of intersection of the number lines as the origin.

31 31

Let 31 be given. The graph of the function f is denoted by Gf and defined as the following set:

31

Example:

31

The graph of the function 31

31

The graph of the piecewise function is denoted by x+1 for x<0 and 1+x/5 for x?0 on 31.