Logics for this lecture:

• Functions \subseteq Relations
• A relation from non-empty set A to non-empty set B is said to be function, if
→ every element of set A has only one image in set B.
In other words, Domain = A and
no distinct ordered pairs have same first element.

DIFFERENT MAPPINGS OF FUNCTIONS

One-One / Injective Mapping / Monomorphism

A function is said to be one-one if each member of the range of the function arises for one and only one member of the domain of the function.
• A function f: A → B is said to be one-one if  f(a) = f(b) \Rightarrow a = b

Many-One

A function is said to be many-one if at least one member of the range of the function arises for more than one member of the domain of the function.
• A function f: A → B is said to be one-one if  f(a) = f(b) \Rightarrow a \ne b

Onto Mapping / Surjective Function

• In a Surjective function,
Range = Co-domain

Into Mapping

• In a Surjective function,
Range \subset Co-domain

• A function which is both one-one and onto is called Bijective Function.

PART - 1

Relations and Functions Lecture 4 (Part 1)
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Question 1. If f : R \to A , given by f(x) = x^2 – 2x + 2 is onto functions, find set A.

Question 2. Is f : R \to R , given by f(x) = | x -1 | one – one? Give reason.

Question 3. f : R \to B , given by f(x) = sin x is onto function, then write the set B.

Question 4. Let A = R – {3} and B = R – {1}. Consider the function f : A \to B defined by f(x) = \left ( \frac{x-2}{x-3} \right) . Is f one-one and onto? Justify your answer.

Question 5. Check the following functions for one-one and onto:
a. f : R \to R, f(x) = \frac{2x – 3}{7}
b. f : R \to R, f(x) = | x + 1 |
c. f : R – {2} \to R, f(x) = \frac{3x – 1}{x – 2}

Question 6. Show that the function f : R \to R defined by f(x) = \frac{x}{x^2 + 1}, \forall x \in R

PART - 2

Lecture-4 Class 12 Maths 1
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Question 7. Let A = [-1, 1] Then, discuss whether the following functions defined on A are one-one, onto or bijective:
a. f(x) = \frac{x}{2}
b. g(x) = | x |
c. h(x) = x| x |
d. k(x) = x^2

Question 8. Let f : N \to N be defined by f(n)=\left \{ \begin{array}{l} {\frac{n + 1}{2}, if n is odd} \\ {\frac{n}{2}, if n is even} \end{array} \right for all n \in N. State whether the function f is bijective. Justify your answer.

Question 9. Let A and B be sets. Show that f : A × B → B × A such that f (a, b) = (b, a) is bijective function.

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