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Functions and graphs

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Functions and graphs

  1. 1. Functions & Graphs by Mrs. Sujata Tapare Prof. Ramkrishna More A.C.S. College, Akurdi
  2. 2. What of a function? <ul><li>A function is a rule such that which takes every element from a set A and maps it to a Unique element in another set B. </li></ul><ul><li>It is a special type of relation R, where for each x in A there is a unique match y in B such that (x, y) is in R. </li></ul><ul><li>Every function is a subset of a relation. </li></ul><ul><li>Every relation is not a subset of a function. </li></ul>
  3. 3. Function terminology R Z f 4.3 4 Domain Co-domain Pre-image of 4 Image of 4.3 f maps R to Z f(4.3)
  4. 4. Even more functions Not a function Range 1 2 3 4 5 “ a” “ bb“ “ cccc” “ dd” “ e” 1 2 3 4 5 a e i o u It is a function
  5. 5. Some Elementary Functions <ul><li>Identity Function : It maps every element of it’s domain to the same element in co-domain. </li></ul><ul><li>Constant Function : It maps every element of it’s domain to a single element in co-domain. </li></ul><ul><li>Polynomial Function : It is given by f(x) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + … + a n x n . </li></ul><ul><li>Rational Function : It is of the form f(x)/g(x), where f(x) & g(x) are polynomial functions. </li></ul>
  6. 6. 5 5 0 -2 Constant function: f(x) = 5 Rational function: f(x) = (4- x 2 )/(x+2) Polynomial function: f(x) = x 2 Identity function: f(x) = x
  7. 7. Functions can be represented in the following ways <ul><li>By showing relationship with the help of arrows between elements of domain and co-domain. </li></ul>If domain and co-domain are By drawing graphs. Infinite Sets It can be represented by means of formula. Finite Sets
  8. 8. One-one functions <ul><li>A function is one-to-one if each element in the co-domain has a unique pre-image </li></ul><ul><li>Note that there can be un-used elements in the co-domain </li></ul>1 2 3 4 5 a e i o A one-to-one function 1 2 3 4 5 a e i o A function that is not one-to-one
  9. 9. One-one functions Function f : R -> R is one-one for any x, y Є R, f(x) = f(y)  x = y. Examples Let f:R  R defined by, f(x) = 3x – 2 f(x)= f(y)  3x – 2 = 3y – 2  x = y Thus, f(x) is one-one Let f:R  R defined by, f(x) = 3x 2 – 2 f(x)= f(y)  3x 2 –2= 3y 2 –2  3x 2 = 3y 2  x = ± y Thus, f(x) is not one-one
  10. 10. Onto functions <ul><li>A function is onto if each element in the co-domain is an image of some pre-image </li></ul><ul><li>Let f: R  R. If for each y in R there exist x in R such that f(x) = y then f is said to be on-to function. </li></ul><ul><li>f: R  R is onto if </li></ul><ul><li>Range of f = Co-domain </li></ul><ul><li>Note that there can be multiply used elements in the co-domain </li></ul>
  11. 11. One-to-one vs. Onto <ul><li>Are the following functions onto, one-to-one, both, or neither? </li></ul>1-to-1, not onto Onto, not 1-to-1 Both 1-to-1 and onto Not a function Neither 1-to-1 nor onto 1 2 3 4 a b c 1 2 3 a b c d 1 2 3 4 a b c d 1 2 3 4 a b c d 1 2 3 4 a b c
  12. 12. Bijections <ul><li>Consider a function that is both one-to-one and onto: </li></ul><ul><li>Such a function is a one-to-one correspondence, or a bijection </li></ul>1 2 3 4 a b c d
  13. 13. Inverse functions R R f 4.3 8.6 Let f(x) = 2x f -1 f(4.3) f -1 (8.6) <ul><li>An inverse function can ONLY be defined on </li></ul><ul><li>a bijective functions. </li></ul>
  14. 14. Compositions of functions g f f ○ g g(a) f(a) (f ○ g)(a) g(a) f(g(a)) a A B C
  15. 15. Compositions of functions <ul><li>Does f(g(x)) = g(f(x))? </li></ul><ul><li>Let f(x) = 2x +3 Let g(x) = 3x +2 </li></ul><ul><li>f(g(x)) = 2(3x +2) + 3 = 6x +7 </li></ul><ul><li>g(f(x)) = 3(2x +3) + 2 = 6x +11 </li></ul><ul><li>Function composition is not commutative! </li></ul>Not equal!
  16. 16. Useful functions <ul><li>Floor:  x  means take the greatest integer less than or equal to the number. For example,  4/9  =0 </li></ul><ul><li>Ceiling:  x  means take the lowest integer greater than or equal to the number. For example,  4/9  =1 </li></ul><ul><li>Round: round(x) = floor(x+0.5) </li></ul><ul><li>For example round(5.9) = 6, round(5.3) = 5 </li></ul>
  17. 17. Floor:  x  -5 -4 -3 -2 -1 0 1 2 3 4 5 5 4 3 2 1 -1 -2 -3 -4 -5
  18. 18. Limit of a Function <ul><li>ε-δ definition of a limit </li></ul><ul><li>For any ε > 0 there exist δ > 0 such that, |f(x) - l| < ε, whenever |x - c| < δ. </li></ul><ul><li>We write it as, </li></ul><ul><ul><li> lim f(x) = l </li></ul></ul><ul><ul><li>x ->c </li></ul></ul>
  19. 19. Continuous Functions <ul><li>A function is said to continuous if </li></ul><ul><ul><li>lim f(x) = l = f(c) Graph </li></ul></ul><ul><ul><li>x ->c </li></ul></ul><ul><ul><li>OR </li></ul></ul><ul><ul><li>lim f(x) = lim f(x) = f(c) Graph </li></ul></ul><ul><ul><li>x ->c - x ->c + </li></ul></ul>
  20. 20. Differentiable Functions <ul><li>f(x) is differentiable at x = c if </li></ul><ul><ul><li>lim f(x) - f(c) = l = f ' (c) </li></ul></ul><ul><ul><li>x->c x - c </li></ul></ul><ul><ul><li>OR </li></ul></ul><ul><ul><li>lim f(x+h)– f(x) = l = f ' (c) </li></ul></ul><ul><ul><li>h->0 h </li></ul></ul><ul><ul><li>Example </li></ul></ul>
  21. 21. o
  22. 22. o
  23. 23. Differentiable Functions <ul><li>Every differentiable function </li></ul><ul><li>is continuous </li></ul><ul><li>But every continuous function can </li></ul><ul><li>not be differentiable </li></ul><ul><li>For example, |x| is continuous </li></ul><ul><li>but not differentiable </li></ul><ul><li>GRAPH </li></ul>
  24. 24. Increasing & Decreasing Functions <ul><li>f(x) is increasing, if f ' (x) > 0 </li></ul><ul><li>f(x) is decreasing, if f ' (x) < 0 </li></ul><ul><li>Example 1 </li></ul><ul><li>Example 2 </li></ul>
  25. 25. Maxima & Minima of the functions <ul><li>f(x) is Maximum </li></ul><ul><li>if f ' (x) = 0 and f '' (x) < 0 </li></ul><ul><li>f(x) is Minimum </li></ul><ul><li>if f ' (x) = 0 and f '' (x) > 0 </li></ul><ul><li>Example </li></ul>
  26. 26. Mean Value Theorems <ul><li>Rolle’s Theorem </li></ul><ul><li>It is a special case of Lagranges </li></ul><ul><li>mean value theorem. </li></ul><ul><li>Lagrange’s Theorem </li></ul><ul><li>Cauchy’s Theorem </li></ul>
  27. 27. f(a) = f(b) f’(c 1 )=0 f’(c 2 ) = 0 c 1 c 2 a 0 b
  28. 28. o c 1 b a c 2

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