Functions and Their Graphs

A correspondence assigning each element of the set A, one and only one element of the set B
is called a function from the set A into the set B. Functions are generally denoted by the lower case letters such as f, g, h. In this case a function defined from the set A to the set B is denoted by f : A → B.

the set A is called the domain or the departure set of the function.

the set B is called the range or terminal set.

The set of all elements composed of the images of each element of A is called the image set of the function.

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 Df
for the rule y=f (x) and is called the natural domain of the function.

If the images of two different elements from the domain under the action of f are also different, such functions are called one-to-one.

Although the range and the image of a function are two concepts that need to be distinguished, we still know that they are closely related. The image is always a subset of the range. Yet, there are functions for which the range and the image sets are exactly the same. Thus, given a function f : A → B if the image is equal to the range, i.e. f (A)=B the function f is called surjective (onto). Equivalently, for every element b of the set B if there can be found an element a of A such that f (a)=b, then f is called onto.

If a function which is both one-to-one and onto is called a bijection.

A function assigning each element from its domain a single element in its range is called a
constant function. That is, for every element a from the domain A, and c∈B, if f (a)=c, the
function f is called constant.

Given A ≠ø, a function defined on the set A and assigning every element of A to itself is called
the identity function.

Just as sets have different representations, functions also have different representations. If
sets have a small number of elements, we may list them. If the elements of sets have a common
property, we denote them by emphasizing this common property.

Functions, which are represented by different formulas on different subsets of its domain are called piecewise defined functions. We frequently encounter piecewise defined functions in our daily lives. For example, suppose that you will be parking your car in a car park. The car park charges 5 T for the first hour. For every following hour, it charges an extra 1,5 T. How much would you pay if you pick your car during the first hour? How much would you pay 1 hour 40 minutes later? How about 2 hours 15 minutes later?
Here, we have a piecewise defined function. We may define this function in the following way: If x
corresponds to the parking time in hours, we have
f :R+ → R

A function which assigns every real number to its distance to the origin, in other words assigning
its absolute value, is called the absolute value function, and it is denoted by | ∙ |. As a piecewise defined function, absolute value is represented as
l⋅l :R → R, x ∈ R

There are many ways of constructing new functions from the given ones. The most important way of constructing a new function from the known ones is the composition of them. We now give the definition of composition. Let the functions f : A→B and g : B→C be given.
The function g∘f : A→C, defined by the rule (g∘f )(a)=g(f (a)) is called the composition of the functions f and g.

The matching of the sets {a, b, c, d } and {1, 2, 3, 4} thus obtained is a function which we call as the inverse function of h. Consequently, having a oneto-one and onto function, it is possible to define a new function by reversing the directions of the arrows as we did in the preceding examples. This newly constructed function is called the inverse of the given function. Mathematically put: Let the bijective function f : A→B be given. The inverse of f is defined as
f –1:B→A, f –1(y)=x

Only bijective functions have an inverse.

Just as the operations of addition, subtraction, multiplication, and division defined on real numbers, it is possible to define similar operations on functions defined from a subset A of real
numbers to real numbers, or a subset of it.

Cartesian coordinate system, a tool which enables us to view the graphs of functions.

The horizontal real line is called the x-axis.

The vertical real line is called the y-axis, or ordinate.

The set of ordered pairs composed of (x, f (x)) for every element x in A is called the graph of the function f :A⊆ R → R .

Let the functions f :A→B and g :B→C be given. You can construct the composition of the functions f and g as g ∘f :A→C, (g ∘f )(x)=g (f (x)), for x ∈A.

The largest domain of the function f(x)= 1/x is the set which makesthe rule meaningfull. f(x)= 1/x
makes sense if its denominator isnot zero, i.e. x ≠ 0 should be satisfied. Thus, the largest domain Df should be taken as R →R\{0}.