Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
Function can be easily defined with the help of the concept of mapping. Let X and Y be any two non-empty sets. "A function from X to Y is a rule or correspondence that assigns to each element of set X, one and only one element of set Y". Let the correspondence be 'f' then mathematically we write f:X → Y
where
y = f(x), x ε X and y ε Y. We say that 'y' is the image of 'x' under 'f' (or x is the pre image of y).
Two things should always be kept in mind:
(i) A mapping f: X → Y is said to be a function if each element in the set X has its image in set Y. It is possible that a few elements in the set Y are present which are not the images of any element in set X.
(ii) Every element in set X should have one and only one image. That means it is impossible to have more than one image for a specific element in set X. Functions can't be multi-valued (A mapping that is multi-valued is called a relation from X to Y)
Set 'X' is called the domain of the function 'f'.
Set 'Y' is called the co-domain of the function 'f'.
Set of images of different elements of set X is called the range of the function 'f'. It is obvious that range could be a subset of co-domain as we may have few elements in co-domain which are not the images of any element of the set X (of course these elements of co-domain will not be included in the range). Range is also called domain of variation.
The set of values for which a function is defined is called the domain of the function. The range of the function is the set of all images of domain of f. In above example, the set A is the domain of the function f. B is not range but the co-domain of the function. The range is the subset of the co-domain. The domain and the range of a function may be an interval, open, closed, semi-closed or semi-open i.e. the domain may be an interval of any of the following types.
If x ε [q, p], then {x : q < x < p}
If x ε (-∞, p], then {x : x < p}
If x ε (-∞, p), then {x : x < p}
If x ε ]-∞, p[, then {x : x < p}
If x ε (p, ∞], then {x : x > p}
If x ε (-∞, ∞), then {x : x ε R}
If x ε {p, q}, then {x : x = p or x q}
Domain of function 'f' is normally represented as Domain (f). Range is represented as Range (f). Note that sometimes domain of the function is not explicitly defined. In these cases domain would mean the set of values of 'x' for which f(x) assumes real values. e.g. if y = f(x) then Domain (f) = {x : f(x) is a real number}.
e.g. Let X = {a, b, c}, Y = {x, y, z}. Suppose f(a) = y, f(a) = x, f(b) = y, f(c) = z. Then f is not a function of X into Y since a ε X has more than one f-images in Y.
On the other hand, if we set f(a) = x, f(b) = x and f(c) = x, then f:X → is a function since each element in X has exactly one f-image in Y.
(i) Let X = R, Y = R and y = f(x) = x2.
Then f : X → Y is a function since each element in X has exactly one f-image in Y. The range of f = {f(x) : x ε X} = {x2 : x ε R} = [0, ∞).
(ii) Let X = R+, Y = R+ and y = √x. Then f : X → Y is a function. The range of f is R+
(iii) Let X = R, Y = R and y2 = x. Her f(x) = +√x i.e. f is not a function of X into Y since each x > 0 has two f-images in Y, and further, each x < 0 has no f-image in Y.
We are primarily interested in functions whose domain and ranges are subsets of real numbers. Such functions are often called Real Valued functions.
e.g. Let the function f be defined by f(x) = 1/√(2x+6).
In this formula we must have 2x + 6 > 0 and therefore x > -3. Therefore, the domain of f is (-3, ∞), the range of f = (0, ∞). Thus we have the function f : (-3, ∞) → (0, ∞) defined by f(x) = 1/√(2x+6).
Let the function f be defined by f(x) = x/((x-1)(x-2)). The formula makes sense for all values of x except x = 1 and x = 2. Therefore, the domain of f is R - {1, 2} IIT JEE study material is available online free of cost at askIITians.com. Study Physics, Chemistry and Mathematics at askIITians website and be a winner. We offer numerous live online courses as well for live online IIT JEE preparation - you do not need to travel anywhere any longer - just sit at your home and study for IIT JEE live online with askIITians.com
To read more, Buy study materials of Set Relations and Functions comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
Composite Functions Problem of finding out fog and...
Bounded and Unbounded Function Let a function be...
Set Theory Table of Content Set Union of Sets...
Graphical Representation of a Function The...
Greatest Integer Function The function f(x) : R...
Graphical Representation of a Function 10....
Constant Function and the Identity Function The...
Absolute Value Function The function defined as:...
Basic Transformations on Graphs Drawing the graph...
Exponential Function Exponential and Logarithmic...
Signum Function The signum function is defined as...
Explicit and Implicit Functions If, in a function...
Periodic Function These are the function, whose...
Even and Odd Function A function f(x) : X → Y...
Logarithmic Function We have observed that y = a x...
Invertible Function Let us define a function y =...
Functions Table of Content What are Functions?...
Increasing or Decreasing Function The function f...
Inverse Function Let f : X → Y be a function...
Set, Relations and Functions – Solved...
Linear Function When the degree of P(x) and Q(x)...
Polynomial and Rational Function A function of the...
Relations Table of Content What do we mean by...
Composite Functions Another useful combination of...
Cartesian Product of Sets Table of Content Define...
Algebra of Functions Given functions f : D →...
Functions: One-One/Many-One/Into/Onto Functions...