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KRISHNA MURTHY IIT ACADEMY

KRISHNA MURTHY IIT ACADEMY. FUNCTIONS By Krishna Murthy. Let X and Y be two nonempty sets of real numbers. A function from X into Y is a rule or a correspondence that associates each element of X with a unique element of Y. The set X is called the domain of the function.

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KRISHNA MURTHY IIT ACADEMY

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  1. KRISHNA MURTHY IIT ACADEMY

  2. FUNCTIONS By Krishna Murthy

  3. Let X and Y be two nonempty sets of real numbers. A function from X into Y is a rule or a correspondence that associates each element of Xwith a unique element of Y. The set X is called the domain of the function. For each element x in X, the corresponding element y in Y is called the image of x. The set of all images of the elements of the domain is called the range of the function.

  4. f x y x y x X Y RANGE DOMAIN

  5. Determine which of the following relations represent functions. Not a function. Function. Function.

  6. Not a function. (2,1) and (2,-9) both work.

  7. Find the domain of the following functions: A) B)

  8. C) Square root is real only for nonnegative numbers.

  9. Theorem Vertical Line Test A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.

  10. y x Not a function.

  11. y x Function.

  12. Domain of a function represents the horizontal spread of its graph & the range the vertical spread. 4 (2, 3) (10, 0) 0 (4, 0) (1, 0) x (0, -3) -4

  13. One-to-One FunctionOn-to FunctionInverse Function

  14. A function f is said to be one-to-one or injective, if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain. A function is not one-to-one if two different elements in the domain correspond to the same element in the range.

  15. x1 y1 x1 y1 x2 y2 x2 x3 y3 x3 y3 Domain Co domain Domain Co domain One-to-one & on-to function NOT One-to-one but on to function One-one but not on to function Domain Co domain

  16. ON-TO or Surjective function A function f is said to be on to if every element of the co-domain is the image of some element of the domain. That is for all y in co-domain, there exist x in domain such that y = f(x). ON-TO ness depends on co-domain

  17. x1 y1 x2 y2 x3 y3 Domain Co domain BIJECTIVE FUNCTION A function is said to be objective if it is both one-one and on-to

  18. Use the graph to determine whether the function is one-to-one. Horizontal line Cuts the graph in more than one point. Not one-to-one.

  19. Use the graph to determine whether the function is one-to-one. One-to-one.

  20. Let f denote a one-to-one function y = f(x). The inverse of f, denoted by f -1 , is a function such that f-1(f( x )) = x for every x in the domain of f and f(f-1(x))=x for every x in the domain of f-1. .

  21. Domain of f Range of f

  22. Theorem The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

  23. y = x (0, 2) (2, 0)

  24. Find the inverse of The function is one-to-one. Interchange variables. Solve for y.

  25. Check.

  26. Even and Odd Functions Even functions are functions for which the left half of the plane looks like the mirror image of the right half of the plane. Odd functions are functions where the left half of the plane looks like the mirror image of the right half of the plane, only upside-down. Mathematically, we say that a function f(x) is even if f(x) = f(-x) and is odd if f(-x) = -f(x).

  27. Some Examples

  28. y = cosx, Even y = sinx, Odd

  29. Is there a function which is both even as well as odd?

  30. Yes there is Only one function which is both even as well as odd

  31. The function is y = f(x) = 0 Let y = f(x) be one such function Then, f(-x) = f(x) and f(-x) = -f(x) So, f(x) = -f(x) f(x) = 0

  32. PERIODIC FUNCTIONS

  33. Periodic functions are functions that repeat over and over, or cycle on a specific period. This is expressed mathematically that A function f is periodic if there exists some number p>0 such that f(x) = f(x+p)for all possible values of x The least possible value of p is called the fundamental period of the function.

  34. f(x) = sinx, is a periodic func with fundamental period 2π f(x) = cosx, is also a periodic func with fundamental period 2π

  35. y = tanx & y = cotx are periodic functions with fundamental period π Graph of y = tanx

  36. A property of some periodic functions that cycle within some definite range is that they have an amplitude in addition to a period. The amplitude of a periodic function is the distance between the highest point and the lowest point, divided by two. For example, sin(x) and cos(x) have amplitudes of 1.

  37. COMBINATIONS OF PERIODIC FUNCTIONS There are no hard and rigid rules for finding the periods of functions which are the combinations of periodic functions but the following technique may work in many cases. If the period of f(x) is (a/b)π and that of g(x) is (c/d)π,then the period of A.f(x) + B.g(x),where A and B are real numbers is (LCM of a,c)/(HCF of b,d) times π

  38. For example, find the period of y = sin7x + tan(5/3)x. Period of sin7x is 2π/7 and that of tan(5/3)x is 3π/5. Hence the period of the given function is (LCM of 2,3)/(HCF of 7,5) times π that is 6 π

  39. If the period of f(x) is p then that of a.f(x) + b is also p and that of f(ax+b) is p/|a| For e.g, period of sin(4-3x) is 2π/3

  40. If f(x) is periodic and g(x) is non periodic then f{g(x)} is not periodic except when g(x) is linear. For e.g, y = sin(4-3x2) is not periodic

  41. A constant function is periodic but has no fundamental period. y = x – [x] is a periodic function whose fundamental period is 1

  42. BEHAVIOR OF FUNCTIONS By behavior of a function, we mean, its Increasing & Decreasing nature Increasing & Decreasing Functions

  43. A function f(x) is said to be increasing in an interval, if for any x1, x2 belonging to this interval, x1 < x2 implies f(x1) ≤ f(x2) OR x1 >x2 implies f(x1) ≥ f(x2) That is, if x increases then f(x) should increase and if x decreases then f(x) should decrease. The function is said to be strictly increasing if x1 < x2 implies f(x1) < f(x2) OR x1 >x2 implies f(x1) > f(x2)

  44. The function y = 3x is strictly increasing

  45. A function f(x) is said to be DECREASING in an interval, if for any x1, x2 belonging to this interval, x1 < x2 implies f(x1) ≥ f(x2) OR x1 > x2 implies f(x1) ≤ f(x2) That is, if x increases then f(x) should decrease and if x decreases then f(x) should increase. The function is said to be strictly decreasing if x1 < x2 implies f(x1) > f(x2) OR x1 >x2 implies f(x1) < f(x2)

  46. The function y = tanx is strictly increasing . The function y = -[x] is decreasing but not strictly decreasing DRAW THE GRAPH and verify.

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