2.2 Functions  (Page 2/3)

$$\text{Domain of "f"}=A$$

$$\text{Co-domain of "f"}=B$$

$$\text{Range of "f"}=\text{Set of images}=\{f\left(x\right):x\in A\}$$

Equal functions

Two functions are equal, if each ordered pair in one of the two functions is uniquely present in other function. It means that if “g” and “h” be two equal functions, then :

$$g\left(x\right)=h\left(x\right)\text{for all}“x”$$

Two functions g(x) and h(x) are equal or identical, if all images of two functions are equal. Further, we can visualize equality of two functions in a negative context. If there exists “x” such that g(x)≠ h(x), then two functions are not equal. We state this symbolically as :

$$\text{If}g\left(x\right)\ne h\left(x\right),\text{then}f\ne h$$

The important question, however, is that whether equality of functions in terms of equality of images is a sufficient condition? We can see here that two functions can meet the stated condition even if they are constituted by different sets of ordered pair. There may be additional ordered pairs, which are present in one, but not in other.

In order to remove such possibilities, two equal functions should have same domain. This will ensure that set of ordered pairs in two functions are same. We conclude this discussion by saying that two functions are equal, iff

• g(x) = h(x) for all “x”
• Domain of “f” = Domain of “h”

It is clear that equality of functions, however, do not require that co-domains be equal.

Real function

If the range of a function is a set of real numbers, then the function is called “real valued function”. In other words, if the range of a function is either the set “R” or its subset, then it is a real valued function. We should emphasize here that “R” denotes set of real number and it is not the symbol for relation, which is also denoted as “R”.

Further, we distinguish “real valued function” from “real function”. The very terminology is indicative of the difference. The term “real valued function” means that the value of function i.e. image is real. It does not say anything about “pre-image”. Now, there can be a function, which accepts non-real complex numbers, but maps to a real value.

On the other hand, a real function has both image and pre-image as real numbers. It follows then that the domain of a “real function” is also either a set or subset of real numbers.

Real function

A function is a real function, if its domain and range are either “R” or subset of “R”.

Interpretation of function relation

It is intuitive to find similarity of an algebraic equation to the “rule” of a function. Consider an equation,

$$y=x^2+1$$

This equation is valid for all real values of “x”. The set of real values of “x”, belongs to set “R”. The set of values of “y” also belongs to set of “R”. On the other hand, the equation itself is the rule that maps two sets comprising of values of “x” and “y”.

Alternatively, we can write the rule also as :

$$\Rightarrow f\left(x\right)=x^2+1$$

In terms of rule, we define function, saying that :

$$f:R\to Rbyf\left(x\right)=x^2+1$$

We read it as : “f” is a function from “R” to “R” by the rule given by $f\left(x\right)=x^2+1$ .

From this description, we think a function as a relation, which is governed by a specified rule. The rule relates two sets known as domain and co-domain, which are sets of real numbers. One of the quantities “x” is independent of other quantity “y”. The other quantity “y” is a dependent on quantity “x”. In plain words, one of the interpretations is that function relates dependent and independent variables. As a matter of fact, we would attach additional meanings to the concept of function as we proceed to study it in details.