Functions

1. Functions Determine Whether a Relation is a function Find the value of a function Find the Domain of the Function Form the Sum, Difference, Product and Quotient of Two Functions

2. An example of a relation A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y then we say that x corresponds to y or that y depends on x. x→y ( y depends on x)

3. Continued The relation “was born on” Me Hugh Billy Liam Erica March 1 November 12 August 10 June 23

4. Definition of Function Let X and Y be two nonempty sets. A function from X into Y is a relation that associates with each element of X exactly one element Y. X is called the domain Y is called the range

5. Any easy way to remember In words For a function, no input has more than one output. For a function the domain is the set of inputs and the range is the set of outputs.

6. Example of a function Look at y=2x-5 1≤ x≤ 6 Is it a function? Why? What is the domain? What is the range? Remember INPUT AND OUTPUT!!

7. Function notation f(x) is the value of f at the number x;

8. Helpful Hint Think of a function as a machine It only accepts numbers from the domain of the function For each input, there is exactly one output (which may be repeated for different inputs)

9. Finding values of a function f(x)= 2x2-3x Find f(3) f(x) + f(3) f(x+3)

10. Implicit Form of a Function When a function f is defined by and equation in x and y, we say that the function f is given implicitly 3x+y =5 If it is possible to solve the equation fo y in terms of x, then we write y= f(x) and say that the function is given explicitly y= f(x) = -3x+5

11. Homework Page 61 examples 15-25 first column examples 27,28 a-h examples 35,39,43

12. Summary from Wednesday • For each x in the domain of f there is exactly one image f(x) in the range; however an element in the range can result from more than one x in the domain • f is the symbol that we use to denote the function .It is symbolic of the equation that we use to get from an x in the domain to f(x) in the range • If y= f(x), then x is called the independent variable or argument of f and y is called the dependent variable or the value of f at x.

13. Finding the domain of a function f(x) = x2+5x f(x) = F(x) =

14. Operations of functions If f and g are functions The sum f + g is the functions defined by (f + g)(x) = f(x) + g(x)

15. The difference The difference f-g is the function defined by (f-g)(x) = f(x) – g(x)

16. The product and the quotient The product f ∙ g (x) = f(x) ∙ g(x) The quotient g(x) ≠ 0

17. Graphs of Functions Identify the graph of a Function Obtain Information from or about the Graph of a Function

18. Graph of IBM stock

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

20. Identifying the Graph of a function

21. More Graphs

22. Obtaining Information from the graph of a function

23. The information that we want from the graph • What are f(0) • What is the domain of f • What is the range of f • List the intercepts. (Recall that these are the points, if any, where the graph crosses or touches the coordinate axes) • How often does the line y=2 intersect the graph? • For what values of x does f(x) = -4 • For what values of x is f(x) > 0

24. Properties of Functions Determine Even and Odd functions from the graph Identify Even and Odd Functions from the Equation Use a graph to Determine where the function is increasing, decreasing or constant Use the graph to locate Local Maxima and Minima Use a graphing utility to approximate local Maxima and Minima and to determine where the function is increasing and decreasing

25. Properties of Functions It is easiest to obtain the graph of a function y = f(x) by knowing certain properties that the function has and the impact of these properties on the way that the graph will look. Intercepts If x= 0 is in the domain of a function y = f(x), then the y intercept of the graph of f is the value of f at 0, which is f(0). The x intercept of the graph of f , if there are any, are the solutions of the equation f(x) = 0 The x intercepts of the graph of a function f are called the zeros of f

26. Even or Odd Functions A function is even if, for every number x in its domain, the number –x is also in the domain and f(-x) = f(x) A function f is odd if, for every number x in its domain, the number –x is also in the domain and f(-x) = -f(x)

27. Theorem A function is even if and only if its graph is symmetric with respect to the y axis. A function is odd if and only if its graph is symmetric with respect to the origin.

28. Determining even and odd functions from the graph

29. Identifying Even and Odd functions Algebraically Determine whether each of the following functions is even, odd, or neither. Then determine whether the graph is symmetric with respect to the y axis or with respect to the origin • f(x)= x2-5 • f(x) = 5x3 –x • f(x) = x3-1 • F(x) = |x|

30. Increasing and Decreasing functions A function f is increasing on an open interval I if, for any choice of x1 and x2 in I with x1<x2 we have f(x1) <f(x2). A function f is decreasing on an open interval I if, for any choice of x2 and x2 in I, with x1<x2 we have f(x1)>f(x2) A function f is constant on an interval I if, for all choices of x in I, the values f(x) are equal

31. Local Maximum and Local Minimum A function f has a local maximum at c if there is an open interval I containing c so that, for all x ≠ c in I f(x) < f(c). We call f(c) a local maximum of f A function f has a local minimum at c if there is an open interval I containing c so that, for all x ≠ c in I, if f(x) > f(c). We call f(c) a local minimum of f.