![Function Function](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFhdRWQJ5RVXJcPEZNbrwHqSX70i9dEfj5OWh5MvNNc9LREDg7zic2mpv_YrtIXPSOnzDpIvFEytL2_eBtwlsylOXvxUJsdY2aGyF8aeOSBfGph4BOHxFzSDRLKYxTGLw0ZuHCNaX1qx7kIXYDFVfTu74KuF9YTVLSbSTBnYU6Zl7k-wAenimRxlWElw/s16000/Function.png)
Consider two sets A and B. Any non-empty subset R of the
cartesian product A × B is called a relation from A to B. A special type of
relation associates each element of set A with the one and only element of B.
This is indeed a refinement of the concept of relation. Such a refinement is
known as a function. We may define a function in the following way:
A function from a set A to a set B is a relation (or rule) which
associates each element of A with a unique element of B.
Symbolically, we write
f : A → B
to mean “f is a function from A to B”. Further, an element y of
B associated with an element x of A is denoted by f(x). Equivalently, we write
y = f(x) which reads ‘y equals f of x’. Here f(x) is known as the image of f at
x or the value of f at x.
The letters f, g, h, F, G, H, φ etc. are reserved for denoting
functions.
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Representation of Functions
The various ways by which a function can be represented are
mentioned below:
(i) A table
(ii) A set of ordered pairs
(iii) An arrow diagram
(iv) A graph
(v) A machine
(vi) A formula
Example: Let A = {1, 2, 3} and B = {1, 4, 9}. And a function assigns
each number of A with its square in B. So, f(1) = 1, f(2) = 4 and f(3) = 9. Here,
the domain of the definition of f is A and the range of f is f(A) = B.
This function f : A → B can be represented by
(a) A table:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggFfT-2GB4hSu72LqUIk_0PNtUaXVurx0LTKbOEzdEwc0Dk84n5QL6s5gOzE-DlnnBmjnsiLn3SLheV6BFITKWJT7jBn4p60N2FvEa4vPfKufxL3wTFGSDZnEL1YkE5qc_NDvQ3MWwmAt7kX0o4_MXqOYSbtCWq6oMLV_VL29N5XMqFdaDy1tBOJobeg/s1600/Table.png)
(b) A set of ordered pairs:
f = {(1, 1), (2, 4), (3, 9)}
(c) An arrow diagram:
![Arrow diagram Arrow diagram](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvQYnj8Fbj5jI45NXrEHRU-9C6Q2Q76u2SQ7PyLmQJ4FHYmXdx51x6F5H5eZl_EcBGRvr6YPK760Z4KxB5gAIGCYF9AegXyMpCScMyYpy-IVEtz0cXDytLW3QogEj4DMBugnkWsKU_-R-TiMCZRP6EkLaIcft0Ufmd5e2HRus9nxTznMWbYUN0URNTTQ/s16000/arrow%20diagram.png)
(d) A graph:
![Graph Graph](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiG-x8B-9C638OYe4Pbu5om2y4cmPeiV82L0fp3UqR1txhzxxLJVNsvLDrfUe8EiQ-YGcZgFwxaHYYfiE6LMcOmy32s8n9RSM4ovFpXe3Y8E0jVy07yvZUGest21hTuZu7Vp7NRlIZzQjCGyyeIPTvByQG2G1UZPy3O5D9V5pGdcc3uzvTzUwh2vgzwfA/s16000/graph.png)
(e) A machine:
![Machine Machine](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlutRTHT2EM2_X6_8MTKWbcZfmNPd8PnGTKTIAilfUNjIQSCwsfQdB2vfct-JMEqkn88ONet1QFFiIYDRuIRCB6tv2MYJMzFpjv5Pdp8c8nHRME9Ak-dZ9LBUDIsMsYVxFvuXEM00VSOfIYHJa1MpW6liooSFlCIfNYeFFm59bUi6CNtnWivD5KmvqvA/s16000/machine.png)
(f) A formula:
y = f(x) = x2, where x ∈ A and y ∈ B.
Testing of Function
We can test whether a given relation is a function or not by
applying the following tests.
(i) If a function is in the form of a set of ordered pairs,
examine whether the first element of all the ordered pairs are different or
not. If not, it is not a function.
![Functions 1. f = {(1, 2), (2, 3), (3, 4)} 2. g = {(4, 5), (6, 7), (8, 9)} 3. h = {(2, 3), (4, 3), (5, 3)} Every x-components are distinct. Not the functions 1. R1 = {(1, 2), (1, 3), (2, 3)} 2. R2 = {(4, 5), (6, 7), (6, 9)} 3. R3 = {(a, 1), (a, 2), (a, 3)} At least two x-components or the ordered pairs are the same.](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBo9ZbK9j0j7JA6ULfYG7W9Qu3TOB1JTxZ_yzjkWV-xZ5EJe--ajamMK-s9ZNNpjHiJpMDZ80hDYfdN3wH9YYt96EfSn9sL9EhjSRdxBsg5v_uNreX2yrNfZnTfm_cr_fnOVn4pW_bxbOab1HyLDhZaPkZ27_QiYfUksNs4TzYYTJl__tLQ340ZTHLMA/s16000/test%20of%20function%201.png)
(ii) In the arrow diagram, examine whether each element of the first set has only one image in the second set or not. If not, it is not a function.
![Functions: Every element of f has unique image. Not the functions: At least one element of R does not have unique image.](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUAz7RqIodubr8tgoMBzNQNAcquMkviM5O1kGbc3o6LKiBIkYOnB88__ahhfHj1Q37GI048zqnBjAV0WPCwmAV0MHiQRZNfiSFOAu-6d-SbGoCQ68iCNRc_sKWqM6z-Qr4z00RdD4nAutpkRfqYCOTU52HK5JDtGwW7Hxdp91ZSdr1xIGJXPpwHa2sew/s16000/test%20of%20function%202.png)
(iii) Vertical Line Test: In the graph of a relation, it will
represent a function if each vertical line cuts the graph at only one point.
The graph will not represent a function if a vertical line cuts the graph at
more than one point.
![Graphs of vertical line test: Functions Vertical line cuts the graph at only one point. Not the functions Vertical line cuts the graph at more than one point.](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkuN1KTTVtUatmb2z7oD3VEWJBI1BHPRMAwAVLaJqa1BRIBFXFHyDV9xYXlUTltTV7_Ltb6cSwu7rBdUNWZh-rTfPwe4P3xvrzy5gMfRBCzlcbyWd0kx0qb4V6f7aa3WUN8kN2NmL25c8rC3mciJGbTQNff8gWMYqn-47SRH1cxxQT-sqGbHcdNKyLZg/s16000/test%20of%20function%203.png)
Domain, Range, and Co-domain of a Function
If f is a function from set A to set B i.e. f : A → B then set A is said to be the domain of f and B, the co-domain of f. The set of
elements of B which are the images of the elements of A is known as the range
of f.
The image of a function is the set of all output values it may
produce. For a given function, the set of all elements of the domain that are
mapped into a given subset of co-domain (Range) is known as the pre-images and
the elements of the range are called images.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEithmwx2IVQTIxlY9cIoevU-TRfhaQ-OorgHZcYlqRsKQC1mGIS-x7Bih_rURGZVFyYP5OnEku_xqF_39j7ZGoKTrvgzdWyDFt5yMtWI2p1-F5o6CryvtzTpAfKaL5qb3Sf5p-nAM16EppRh5h-ekHEALiAQOwbfp8d-GB8i7BkVGSIgVDblXiy37OFgg/s320/domain%20co-domain.png)
Types of Functions
Onto
Function:
A function f is called an onto function if its range and
co-domain are equal.
Into
Function:
A function f is said to be an into function if its range is a
proper subset of its co-domain.
One to
one Function:
A function f is called one to one function if every element in
the range has a single pre-image.
Many to
one Function:
A function f is called many to one function if at least one
element of the range has more than one pre-image.
Value of a Function
If f is a function and (p, q) is in f then we write f(p) = q,
where f(p) is called the value of the function at p.
Worked Out Examples
Example 1: Let A = {1, 2, 3}, B = {2, 4, 6} and f : A → B such that f(1) = 2,
f(2) = 4 and f(3) = 6. Represent the function f by
(i) Tabular form
(ii) Set of ordered pairs
(iii) Arrow diagram
(iv) Graph
(v) Formula (an equation)
Solution: Here,
A = {1, 2, 3}
B = {2, 4, 6}
f : A → B
(i) In a tabular form:
f(1) = 2 i.e. when x
= 1 then y = 2
f(2) = 4 i.e. when x
= 2 then y = 4
f(3) = 6 i.e. when x
= 3 then y = 6
![Table Table](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhal9s6HxlaYhasJMruihpwFXPlQXM87V6Z8SnlQ2lkugqNr2TAZdGAm_ertOYGknXs5ztNBvHHwhUEI7OZOYb29hFlBz26d_KE6-Hy0AqGJmopQ2BFLEZxqQXIiwEklx8YWYH38LGnRwNOwqZh9dqHfmNY2H0q10CncQgImg8ObRBY251NQU-O3Dgs1g/s16000/Example%201%20Table.png)
(ii) A set of ordered pairs form:
f = {(1, 2), (2, 4), (3, 6)}
(iii) In an arrow diagram:
1 corresponds with f(1) = 2
2 corresponds with f(2) = 4
3 corresponds with f(3) = 6
![Arrow diagram Arrow diagram](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibGoqhPMVs0_ywkxyvEGDn36GH8_2GPQxWS59eyPstCdFVzcZssYk_jwulhNaJmGYbpU5G6owDQVIAj0joc09k_UBkez2XgjasXZVi4j0jwswO2Xj5j3Hh5rvuiD145gAfOev5JVxBkQ3Iq3zHsU4RiOac7zL3r997dM9BmQ2ZwdEApeoQdPIZ5q0j5w/s16000/Example%201%20Arrow%20Diagram.png)
(iv) In a graph:
The points with ordered pairs: (1, 2), (2, 4), (3, 6) are plotted.
![Graph Graph](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqHw-ICFVz3XpyQGpLmc4k5bJHp7R6lOaIWQ17P-m4SEaY1p0q6zPJp_2G7VrnOX7pOpDlnWHiqf4hs18hQOUzQ_3rJC2bxzz9auMnpIVGWKGzabf4byIpgbjX2EHN-A8Hb7Lad3hjMxbUggI-5JrAtKfXRwQ0C6AydNPZ6o_wahyDdpgQaQSUAspPkg/s16000/Example%201%20Graph.png)
(v) In formula (an equation):
When x = 1 then y = f(1) = 2 = 2 × 1
When x = 2 then y = f(2) = 4 = 2 × 2
When x = 3 then y = f(3) = 6 = 2 × 3
∴ The function f can be expressed as y = 2x or f(x) = 2x.
Example 2: What element in the domain has image 9 under the function f(x) = 4x
+ 5?
Solution: Here,
f(x) = 4x + 5 and image of f = 9
So, f(x) = 9
or, 4x + 5 = 9
or, 4x = 9 – 5
or, 4x = 4
or, x = 4/4
or, x = 1
Thus, the required element of the domain is 1.
Example 3: If a function f is such that f(x + 5) = f(x) + f(5), x ∈ R, show that f(0) = 0 and f(-5) = - f(5).
Solution: Here,
f(x + 5) = f(x) + f(5)
at x = 0
f(0 + 5) = f(0) + f(5)
or, f(5) = f(0) + f(5)
or, f(5) – f(5) = f(0)
or, 0 = f(0)
∴ f(0) = 0
Again, at x = -5
f(-5 + 5) = f(-5) + f(5)
or, f(0) = f(-5) + f(5)
or, 0 = f(-5) + f(5)
or, -f(5) = f(-5)
∴ f(-5) = -f(5) proved.
Example 4: A function f is defined by f(x) = x2 + 4x + 5 then find
the value of
(i) f(5)
(ii) f(-2)
(iii) f(8) + f(9)
Solution: Here,
f(x) = x2 + 4x + 5
(i) When x = 5 then, f(5) = 52 + 4×5 + 5 = 25 + 20 +
5 = 50
(ii) When x = -2 then f(-2) = (-2)2 + 4 × (-2) + 5 =
4 – 8 + 5 = 1
(iii) When x = 8 then f(8) = 82 + 4×8 + 5 = 64 + 32 +
5 = 101
When x = 9 then f(9) = 92 + 4×9 + 5 = 81 + 36 + 5 =
122
Now, f(8) + f(9) = 101 + 122 = 223
Example 5: Let A = {a, b, c}, B = {5, 7, 10, 12} and define f(a) = 5, f(b) =
7, f(c) = 10. Identify whether function f : A → B is into or onto.
Solution: Here,
f(a) = 5, f(b) = 7 and f(c) = 10
Range = {5, 7, 10}
Co-domain = {5, 7, 10, 12}
The range is a proper subset of the co-domain. Thus the given function
is an into function.
Example 6: Let the function f : N → N be defined by f(x) = 2x + 1
(i) Is the function f onto?
(ii) What is the range of 8?
(iii) What is the pre-image of 11?
(iv) If (k, 4k – 5) lies in the function, find the value of k.
Solution: Here,
(i)
Domain = {1, 2, 3, …………..}
Range = {1, 2, 3, …………….}
y = f(x) = 2x + 1
When x = 1 then y = 2×1 + 1 = 3
When x = 2 then y = 2×2 + 1 = 5
When x = 3 then y = 2×3 + 1 = 7
∴ Range = {3, 5, 7, ………….} ≠ Co-domain of f
Thus, f is not an onto function.
(ii)
When x = 8 then y = 2×8 + 1 = 17
Thus, the image of 8 is 17.
(iii)
For pre-image of 11, f(x) = 11
or, 2x + 1 = 11
or, 2x = 11 – 1
or, x = 10/2
or, x = 5
∴ The pre-image of 11 is 5.
(iv)
(k, 4k – 5) lies on f(x) = 2x + 1
So, x = k and f(x) = 4k – 5
Now, f(x) = 2x + 1
or, 4k – 5 = 2k + 1
or, 4k – 2k = 1 + 5
or, 2k = 6
or, k = 3
∴ The value of k is 3.
Example 7: If f(x) and g(x) be two functions defined by f(x) = 4x2
– 3x + 4 and g(x) = 3x2 – 7x + 9 such that f(x) = g(x). Find the
value of x.
Solution: Here,
f(x) = g(x)
or, 4x2 – 3x + 4 = 3x2 – 7x + 9
or, 4x2 – 3x + 4 – 3x2 + 7x – 9 = 0
or, x2 + 4x – 5 = 0
or, x2 + 5x – x – 5 = 0
or, x(x + 5) – 1(x + 5) = 0
or, (x + 5)(x – 1) = 0
∴ Either: x + 5 = 0 or x = -5
Or: x – 1 = 0 or, x = 1
∴ x = -5, 1
Thus, the value of x are -5 and 1.
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