**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

**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

**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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