Mathematics

1. Mathematics

2. Session Functions, Limits and Continuity-1

3. Session Objectives • Function • Domain and Range • Some StandardReal Functions • Algebraof Real Functions • Even and Odd Functions • Limit of a Function; Left Hand and Right Hand Limit • Algebraic Limits : Substitution Method, Factorisation Method, Rationalization Method • Standard Result

4. If f is a function from a set A to a set B, we represent it by If f associates then we say that y is the image of the element x under the function or mapping and we write Function If A and B are two non-empty sets, then a rule which associates each element of A with a unique element of B is called a function from a set A to a set B. Real Functions: Functions whose co-domain, is a subset of R are called real functions.

5. The set A is called the domain of the function and the set B is called co-domain. Domain and Range The set of the images of all the elements under the mapping or function f is called the range of the function f and represented by f(A).

6. Domain and Range (Cont.) For example: Consider a function f from the set of natural numbers N to the set of natural numbers N i.e. f : N N given by f(x) = x2 Domain is the set N itself as the function is defined for all values of N. Range is the set of squares of all natural numbers. Range = {1, 4, 9, 16 . . . }

7. Example– 1 Find the domain of the following functions:

8. Example– 1 (ii) The function f(x) is not defined for the values of x for which the denominator becomes zero Hence, domain of f = R – {1, 2}

9. Example- 2 Find the range of the following functions:

10. Example – 2(ii) • -1  cos2x  1 for all xR • -3  3cos2x  3 for all xR • -2  1 + 3cos2x  4 for all xR •  -2 f(x)  4 • Hence , range of f = [-2, 4]

11. Some Standard Real Functions (Constant Function) Y f(x) = c (0, c) X O Domain = R Range = {c}

12. Identity Function Y I(x) = x 450 X O Domain = R Range = R

13. Modulus Function Y f(x) = x f(x) = - x X O

14. Example y = sinx y = |sinx|

15. Greatest Integer Function = greatest integer less than or equal to x.

16. Multiplication by a scalar: For any real number k, the function kf is defined by Algebra of Real Functions

17. Algebra of Real Functions (Cont.)

18. Composition of Two Functions

19. Example - 3 Let f : R R+ such that f(x) = ex and g(x) : R+R such that g(x) = log x, then find (i) (f+g)(1) (ii) (fg)(1) (iii) (3f)(1) (iv) (fog)(1) (v) (gof)(1) Solution : (i) (f+g)(1) (ii) (fg)(1) (iii) (3f)(1) = f(1) + g(1) =f(1)g(1) =3 f(1) = e1 + log(1) =e1log(1) =3 e1 = e + 0 = e x 0 =3 e = e = 0 (iv) (fog)(1) (v) (gof)(1) = f(g(1)) = g(f(1)) = f(log1) = g(e1) = f(0) = g(e) = e0 = log(e) =1 = 1

20. Example – 4 Find fog and gof if f : R  R such that f(x) = [x] and g : R  [-1, 1] such that g(x) = sinx. Solution:We have f(x)= [x] and g(x) = sinx fog(x) = f(g(x)) = f(sinx) = [sin x] gof(x) = g(f(x)) = g ([x]) = sin [x]

21. Even and Odd Functions Even Function : If f(-x) = f(x) for all x, then f(x) is called an even function. Example: f(x)= cosx Odd Function : If f(-x)= - f(x) for all x, then f(x) is called an odd function. Example: f(x)= sinx

22. Prove that is an even function. Example – 5

23. Example - 6 Let the function f be f(x) = x3 - kx2 + 2x, xR, then find k such that f is an odd function. Solution: The function f would be an odd function if f(-x) = - f(x)  (- x)3 - k(- x)2 + 2(- x) = - (x3 - kx2 + 2x) for all xR  -x3 - kx2 - 2x = - x3 + kx2 - 2x for all xR • 2kx2 = 0 for all xR • k = 0

24. x 2.5 2.6 2.7 2.8 2.9 2.99 3.01 3.1 3.2 3.3 3.4 3.5 f(x) 5.5 5.6 5.7 5.8 5.9 5.99 6.01 6.1 6.2 6.3 6.4 6.5 Limit of a Function As x approaches 3 from left hand side of the number line, f(x) increases and becomes close to 6

25. Limit of a Function (Cont.) Similarly, as x approaches 3 from right hand side of the number line, f(x) decreases and becomes close to 6

26. Left Hand Limit Y x takes the values 2.91 2.95 2.9991 .. 2.9999 ……. 9221 etc. x X O 3

27. Right Hand Limit Y 3 X O x x takes the values 3.1 3.002 3.000005 …….. 3.00000000000257 etc.

28. Existence Theorem on Limits

29. Example – 7 Which of the following limits exist:

30. Example - 7 (ii)

31. Properties of Limits If and where ‘m’ and ‘n’ are real and finite then

32. Algebraic Limits (Substitution Method) The limit can be found directly by substituting the value of x.

33. When we substitute the value of x in the rational expression it takes the form Algebraic Limits (Factorization Method)

34. When we substitute the value of x in the rational expression it takes the form Algebraic Limits (Rationalization Method)

35. Standard Result If n is any rational number, then

36. Example – 8 (i)

37. Example – 8 (ii)

38. Example – 8 (iii)

39. Solution Cont.

40. Example – 8 (iv)

41. Example – 8 (v)

42. Thank you