1. (MTH 250) Lecture 2 Calculus

2. Previous Lecture’s Summary Introduction. Purpose of Calculus. Axioms of Order. Absolute value. Archimedean Property. Axioms of Archimedean Field. Axiom of Completeness. Completeness of R.

3. Today's Lecture Functions Graph of functions Domain & range of functions Piecewise defined functions Increasing & decreasing functions Maxima & Minima Even & odd functions Translation, reflection and dilatation

4. Functions A function f from set A to set B is a rule of correspondence that assigns to each element x in set A exactly one element y in set B. A B A is called the domain (or set of allowable inputs) of f B is called the co-domain of f and contains Range (or set of outputs) of f. f x y =f(x)

5. Each element in A (domain) must be matched with an element of B (range). Each element in A is matched to not more than one element in B. Some elements in B may not be matched with any element in A. Two or more elements of A may be matched with the same element of B. Cont..

6. Cont Cont. Functions x Domain y1y2y1y2 Range f y f x1x1 Domain x2x2

7. Not a function Cont.. x Domain y1y2y1y2 Range f y f x1x1 Domain x2x2

8. Cont...For a given input x, the output of a function f is called the value of f at x or the image of x under f. Sometimes we write y= f(x). The input variable x is called the independent variable and the output y is called the dependent variable. For the function f(x) = 3x - 4, f(5) = 3(5) - 4 = 15 - 4 = 11, f(-2) = 3(-2) - 4 = - 6 - 4 = -10. We say, f of 5 is 11, and, f of -2 is -10.

9. Graph of Functions We call a function f as real-valued function of a real variable, if both independent and dependent variables are real numbers. The graph of a real-valued function f in the xy- plane is defined by the graph of equation y=f(x). The points on the graph of the function f(x) are of the form (x, f(x)) or (x, y) where y=f(x).

10. Cont.. Graph of y = |x| 22 11 00 1 11 2 22 |x||x|x

11. Cont.. Graph of y=x² 4949 3 11 00 1 11 4 22 y=x²x 2

12. Cont.. Graph of 4848 3 21 10 0.5 11 0.25 22 x 2

13. Cont.. Vertical line test: A curve in xy-plane is the graph of some function f if and only if no vertical line intersects the curve more than once. For example following curve is a function

14. Domain & range of functions Generally, the domain is implied to be the set of all real numbers that yield a real number functional value (in the range). Some restrictions to domain: 1. Denominator cannot equal zero. 2. Radicand must be greater than or equal to zero. 3. Practical problems may limit domain.

15. Cont.. Finding the Range of a function: Draw a graph of the function for its given domain. The range is the set of values on the y-axis for which a horizontal line drawn through that point would cut the graph.

16. Cont.. f(x) = (x-2) 2 +3, x  R The Range is f(x) ≥ 3 Range Domain

17. Cont.. f(x) = 3 – 2 x, x  R The Range is f(x) < 3.

18. Cont..

19. Piecewise defined functions A piecewise defined function is a function which assumes different values in different pieces (or parts). For example

20. Cont..

21. Cont.. f(x) =  3, for x  0 f(x)=  3+ x 2, for 0< x  2

22. Increasing & Decreasing Functions Increasing: Graph goes “up” as you move from left to right. Decreasing: Graph goes “down” as you move from left to right. Constant: Graph remains horizontal as you move from left to right.

23. Cont.. f(x) =  3, for x  0 f(x)=  3+ x 2, for 0< x  2

24. Cont..

25. Cont.. Exercise: Find the interval where y is increasing, decreasing or constant.

26. Maxima & minima Relative maximum is the point x where the function y= f(x) assume the maximum value over a given interval (has a peak) Relative minimum is the point x where the function y= f(x) assume its minimum value over a given interval (has a Valley) Global maximum is the point x where the function y= f(x) assume its overall maximum value through the domain. Global minimum is the point x where the function y= f(x) assume its overall minimum value through the domain.

27. Maxima & minima

28. Even & Odd functions Even function: If you plug in –x instead of x in a function f(x) and you get the original function, then it’s even, i.e. f(-x) = f(x) Odd function: If you plug in –x instead of x in a function f(x) and you get negative of the original function, then it’s odd, i.e. f(-x) = -f(x).

29. Translation, Reflection & Dilatation Vertical Shifts: If k is a real number and f(x) is a function, the graph of y = f(x) + k is that of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward. Horizontal Shifts: If h is a real number and f(x) is a function, we say that the graph of y = f(x - h) is the graph of f(x) shifted horizontally by h units. If h 0, then the shift is left.

30. Cont.. (vertical translations) Graph y = |x| 22 11 00 1 11 2 22 |x||x|x

31. Cont.. (vertical translations) Graph y = |x| - 1 122 011 00 01 11 12 22 |x| -1|x||x|x

32. Cont.. (vertical translations) Graph y = |x| +2 422 311 200 31 11 42 22 |x| +2|x||x|x

33. Cont.. (vertical translations) y = 3x 2 y = 3x 2 – 3 y = 3x 2 + 2

34. Cont.. (horizontal translations) y = 3x 2 y = 3(x+2) 2 y = 3(x-2) 2

35. Cont.. (summary of translations) for c>0 To Graph:Shift the Graph of y = f(x) by c units y = f(x) + cUP y = f(x) - cDOWN y = f(x + c)LEFT y = f(x - c)RIGHT

36. Cont.. (reflections) Reflection across the x-axis: The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x). Reflection across the y-axis: The graph of the function, y = f(-x) is the reflection of the graph of the function y = f(x).

37. Cont.. (reflections across x-axis) f(x) = x 3 f(x) = -x 3

38. Cont.. (reflections across y-axis) f(x) = x 3 f(-x) = (-x) 3 = -x 3

39. Cont.. (stretching and shrinking) Vertical Stretching and Shrinking: The graph of y = af(x) is obtained from the graph of y = f(x) by 1. stretching the graph of y = f ( x) by a when a > 1, or 2. shrinking the graph of y = f ( x) by a when 0 < a < 1.

40. Cont.. (vertical stretching & shrinking) y = |x| y = 0.5|x| y = 3|x|

41. Lecture Summary Functions Graph of functions Domain & range of functions Piecewise defined functions Increasing & decreasing functions Maxima & Minima Even & odd functions Translation, reflection and dilatation