Types of Functions
• Let A and B be two non empty sets. Then a function ‘f’ from set A to set B is a rule or method that associates the elements of set A to the elements of set B such that:
 All the elements of set A are associated with the elements of set B.
 An element of set A is associated with a unique element in set B.
• If the images of the distinct elements of A are distinct, then the function f is known as oneone function.
• If the images of the distinct elements of A are the same, then the function f is known as manyone function.
• If for every y belongs to B, there exists an element x in A such that f(x) = y, then f is known as onto function.
• A function f from A to B is called into iff there exists at least one element in B, which is not the image of any element of A.
• A function f : A → B is bijective or one  one onto if it is both one  one and onto.
• A function is said to be surjective function if it is onto function.
• The composition of functions f : A → B and g : B → C is the function
gof : A → C given by gof(x) = g(f(x)).
• If f : A → B and g : B → C are oneone, then gof : A → C is also oneone.
• If f : A → B and g : B → C are onto, then gof : A → C is also onto.
• If f : A → B; g : B → C, are two functions such that gof : A → C is defined and oneone, then both f and g are not necessarily oneone.
• If f : A → B is a function and IA, IB are identity functions on A, B respectively, then
(i) foIA = f (ii) IBof = f
•
• A function f : X → Y is defined to be invertible, if there exists a function
g : Y → X such that gof = Ix and fog = Iy.
• Inverse of a bijective function is unique.
• The inverse of a bijective function f is also bijective and (f–1)–1 = f.
• If f : A → B is bijective, then (i) f–1of = IA (ii) fof–1 = IB
• Let A be a nonempty set and f : A → A, g : A → A be two functions such that
gof = IA = fog , then f and g are both bijectives and g = f–1.
Keywords: Function, Types of functions, Composition of functions, Oneone function, Manyone function, Onto function, Into function, Oneone onto function, Invertible functions, Properties of Invertible Functions, Properties of composition of functions
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Q1
Let X = {1, 2, 3, 4}. A function is defined from X to N as R = {(x, f(x)): x $\in $X, f(x) = ^{x}P_{x  1}}. Then the range of f is
Marks:1Answer:
{1, 2, 6, 24}
Explanation:
$f\left(x\right){=}^{x}{P}_{X1}=\frac{X!}{\left(XX+1\right)!}=X![{}^{n}{P}_{r}=\frac{n!}{\left(nr\right)!}]$
$\therefore f\left(1\right)=1!=1,f\left(2\right)=2!=2,f\left(3\right)=3!=3\times 2\times 1=6$
$f\left(4\right)=4!=4\times 3\times 2\times 1=24$
$\therefore Rangeoff=\{1,2,6,24\}$

Q2
If f(x) = ax + b and g(x) = cx + d, then f[g(x)] – g[f(x)] is equivalent to
Marks:1Answer:
f(d) – g(b).
Explanation:
f(g(x)–g(f(x)) = a(cx + d) + b – {c(ax + b) + d}
= ad + b – bc – d = (ad + b) – (bc + d)
= f(d) – g(b)

Q3Marks:1
Answer:
d = –a
Explanation:

Q4
If f : R R and g : R R defined by f(x) = 2x + 3 and g(x) = x^{2} + 7, then the value of x for which f(g(x)) = 25 will be
Marks:1Answer:
2.
Explanation:
f(g(x)) = f(x^{2} + 7) = 2(x^{2} + 7) + 3 = 25
2x^{2} = 8 x = 2

Q5Marks:1
Answer:
f^{1}(x) = f(x).
Explanation: