1. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §2.1 Intro to Functions

2. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §1.6 → Exponent Rules & Properties  Any QUESTIONS About HomeWork §1.6 → HW-02 1.6 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 3 Bruce Mayer, PE Chabot College Mathematics Ordered Pair Defined  An ordered pair (a, b) is said to satisfy an equation with variables a and b if, when a is substituted for x and b is substituted for y in the equation, the resulting statement is true.  An ordered pair that satisfies an equation is called a solution of the equation

4. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 4 Bruce Mayer, PE Chabot College Mathematics Ordered Pair Dependency  Frequently, the numerical values of the variable y can be determined by assigning appropriate values to the variable x. For this reason, y is sometimes referred to as the dependent variable and x as the independent variable. i.e., if we KNOW x, we can CALCULATE y

5. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 5 Bruce Mayer, PE Chabot College Mathematics Mathematical RELATION  Any set of ordered pairs is called a relation. The set of all first components is called the domain of the relation, and the set of all SECOND components is called the RANGE of the relation

6. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  Find the Domain and Range of the relation: { (Titanic, $600.8), (Star Wars IV, $461.0), (Shrek 2, $441.2), (E.T., $435.1), (Star Wars I, $431.1), (Spider-Man, $403.7)}  SOLUTION The DOMAIN is the set of all first components, or {Titanic, Star Wars IV, Shrek 2, E.T., Star Wars I, Spider-Man}

7. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  Find the Domain and Range for the relation: { (Titanic, $600.8), (Star Wars IV, $461.0), (Shrek 2, $441.2), (E.T., $435.1), (Star Wars I, $431.1), (Spider-Man, $403.7)}  SOLUTION The RANGE is the set of all second components, or {$600.8, $461.0, $441.2, $435.1, $431.1, $403.7)}.

8. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 8 Bruce Mayer, PE Chabot College Mathematics FUNCTION Defined  A function which “takes” a set X to a set Y is a relation in which each element of X corresponds to ONE, and ONLY ONE, element of Y.

9. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 9 Bruce Mayer, PE Chabot College Mathematics Functional Correspondence  A relation may be defined by a correspondence diagram, in which an arrow points from each domain element to the element or elements in the range that correspond to it.

10. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Is Relation a Fcn?  Determine whether the relations that follow are functions. The domain of each relation is the family consisting of Malcolm (father), Maria (mother), Ellen (daughter), and Duane (son). 1.For the relation defined by the following diagram, the range consists of the ages of the four family members, and each family member corresponds to that family member’s age.

11. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Is Relation a Fcn?

12. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Is Relation a Fcn? 1.SOLUTION: The relation IS a FUNCTION, because each element in the domain corresponds to exactly ONE element in the range. For a function, it IS permissible for the same range element to correspond to different domain elements. The set of ordered pairs that define this relation is {(Malcolm, 36), (Maria, 32), (Ellen, 11), (Duane, 11)}.

13. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Is Relation a Fcn? 2.For the relation defined by the diagram on the next slide, the range consists of the family’s home phone number, the office phone numbers for both Malcolm and Maria, and the cell phone number for Maria. Each family member corresponds to all phone numbers at which that family member can be reached.

14. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Is Relation a Fcn?

15. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Is Relation a Fcn? 2.SOLUTION: The relation is NOT a function, because more than one range element corresponds to the same domain element. For example, both an office ph. number and a home ph. number correspond to Malcolm. The set of ordered pairs that define this relation is {(Malcolm, 220-307-4112), (Malcolm, 220-527-6277 ), (MARIA, 220- 527-6277), (MARIA, 220-416-5204), (MARIA, 220-433-8195), (Ellen, 220-527- 6277), (Duane, 220-527-6277)}.

16. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 16 Bruce Mayer, PE Chabot College Mathematics Function Notation  Typically use single letters such as f, F, g, G, h, H, and so on as the name of a function.  For each x in the domain of f, there corresponds a unique y in its range. The number y is denoted by f(x) read as “f of x” or “f at x”.  We call f(x) the value of f at the number x and say that f assigns the f(x) value to y. Since the value of y depends on the given value of x, y is called the dependent variable and x is called the independent variable.

17. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 17 Bruce Mayer, PE Chabot College Mathematics Function Forms  Functions can be described by: A Table A Graph

18. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 18 Bruce Mayer, PE Chabot College Mathematics Function Forms  Functions are MOST OFTEN described by: An EQUATION  NOTE: f(x) ≠ “f times x” f(x) indicates EVALUATION of the function AT the INDEPENDENT variable-value of x

19. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 19 Bruce Mayer, PE Chabot College Mathematics Evaluating a Function  Let g be the function defined by the equation  y = g(x) = x 2 – 6x + 8  Evaluate each function value:  SOLUTION

20. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 20 Bruce Mayer, PE Chabot College Mathematics Evaluating a Function  Evaluate fcn  y = g(x) = x 2 – 6x + 8  SOLUTION

21. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 21 Bruce Mayer, PE Chabot College Mathematics Evaluating a Function  Evaluate fcn  y = g(x) = x 2 – 6x + 8  SOLUTION

22. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  is an EQN a FCN??  Determine whether each equation determines y as a function of x. a. 6x 2 – 3y = 12 b. y 2 – x 2 = 4  SOLUTION a. aany value of x corresponds to ONE value of y so it DOES define y as a function of x

23. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  is an EQN a FCN??  Determine whether each equation determines y as a function of x. a. 6x 2 – 3y = 12 b. y 2 – x 2 = 4  SOLUTION b. TTWO values of y correspond to the same value of x so the expression does NOT define y as a function of x.

24. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 24 Bruce Mayer, PE Chabot College Mathematics Implicit Domain  If the domain of a function that is defined by an equation is not explicitly specified, then we take the domain of the function to be the LARGEST SET OF REAL NUMBERS that result in REAL NUMBERS AS OUTPUTS. i.e., DEFAULT Domain is all x’s that produce VALID Functional RESULTS

25. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Find the Domain  Find the DOMAIN of each function.  SOLUTION a.f is not defined when the denominator is 0. 1−x 2 ≠ 0 → Domain: {x|x ≠ −1 and x ≠ 1}

26. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Find the Domain  SOLUTION The square root of a negative number is not a real number and is thus excluded from the domain x NONnegative → Domain: {x|x ≥ 0}, [0, ∞)

27. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Find the Domain  SOLUTION The square root of a negative number is not a real number and is excluded from the domain, so x − 1 ≥ 0. Thus have x ≥ 1 However, the denominator must ≠ 0, and it does = 0 when x = 1. So x = 1 must be excluded from the domain as well DeNom NONnegative-&-NONzero → Domain: {x|x > 1}, (1, ∞)

28. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Find the Domain  SOLUTION Any real number substituted for t yields a unique real number. NO UNDefinition → Domain: {t|t is a real number}, or (−∞, ∞)

29. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 29 Bruce Mayer, PE Chabot College Mathematics Function Equality  Two functions f and g are equal if and only if: 1.f and g have the same domain and 2.f(x) = g(x) for all x in the domain.

30. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 30 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §2.1 Exercise Set 18, 26  P2.1-26 Functional Relationships

31. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 31 Bruce Mayer, PE Chabot College Mathematics All Done for Today Some Statin Drugs

32. BMayer@ChabotCollege.edu MTH55_Lec-04_Sec_2-1_Fcn_Intro.ppt 32 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –