1. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §2.3 Algebra of Functions

2. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §2.2 → Function Graphs  Any QUESTIONS About HomeWork §2.2 → HW-04 2.2 MTH 55

3. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 3 Bruce Mayer, PE Chabot College Mathematics Function ReVisited  A FUNCTION is a special kind of Correspondence between two sets. The first set is called the Domain. The second set is called the Range. For any member of the domain, there is EXACTLY ONE member of the range to which it corresponds. This kind of correspondence is called a function DomainRange Correspondence

4. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 4 Bruce Mayer, PE Chabot College Mathematics Function Analogy → Machinery  The function pictured has been named f. Here x represents an arbitrary input, and f(x) (read “f of x,” “f at x,” or “the value of “f at x”) represents the corresponding output.

5. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 5 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

6. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 6 Bruce Mayer, PE Chabot College Mathematics Example  Find Function Domain  Find the domain of the Function  First determine if there is/are any number(s) x for which the function cannot be computed?”  Recall that an expression is meaningless for Division by Zero  So In this case the Fcn CanNot be computed when x − 8 = 0

7. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 7 Bruce Mayer, PE Chabot College Mathematics Example  Find Function Domain  This Fcn Undefined for x − 8 = 0  To determine what x-value would cause x − 8 to be 0, we solve the equation:  Thus 8 is not in the domain of f, whereas all other real numbers are.  Then the domain of f is

8. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 8 Bruce Mayer, PE Chabot College Mathematics Algebra of Functions  The Sum, Difference, Product, or Quotient of Two Functions  Suppose that a is in the domain of two functions, f and g. The input a is paired with f(a) by f and with g(a) by g.  The outputs can then be added to obtain: f(a) + g(a).

9. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 9 Bruce Mayer, PE Chabot College Mathematics Algebra of Functions  If f and g are functions and x is in the domain of both functions, then the “Algebra” for the two functions:

10. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 10 Bruce Mayer, PE Chabot College Mathematics Example  Function Algebra  Find the Following for these Functions a) (f + g)(4)b) (f − g)(x) c) (f/g)(x)d) (fg)(−1)  SOLUTION a)Since f(4) = −8 and g(4) = 13, we have (f + g)(4) = f(4) + g(4) = −8 + 13 = 5.

11. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 11 Bruce Mayer, PE Chabot College Mathematics Example  Function Algebra c)(f/g)(x) → Assumes b)(f − g)(x) →  SOLUTION for

12. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 12 Bruce Mayer, PE Chabot College Mathematics Example  Function Algebra  SOLUTION for d)(fg)(−1) → f(−1) = −3 and g(−1) = −2, so

13. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 13 Bruce Mayer, PE Chabot College Mathematics Example  Function Algebra  Given f(x) = x 2 + 2 and g(x) = x − 3, find each of the following. a) The domain of f + g, f − g, fg, and f/g b) (f − g)(x)c) (f/g)(x)  SOLUTION a) The domain of f is the set of all real numbers. The domain of g is also the set of all real numbers. The domains of f +g, f − g, and fg are the set of numbers in the intersection of the domains; i.e., the set of numbers in both domains, or all real No.s For f/g, we must exclude 3, since g(3) = 0

14. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 14 Bruce Mayer, PE Chabot College Mathematics Example  Function Algebra  SOLUTION b) → (f − g)(x) (f − g)(x) = f(x) − g(x) = (x 2 + 2) − (x − 3) = x 2 − x + 5  SOLUTION c) → (f/g)(x) Remember to add the restriction that x ≠ 3, since 3 is not in the domain of (f/g)(x)

15. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 15 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §2.3 Exercise Set 56 by PPT, 10, 30, 36, 42, 64  Demographers use birth and death rates to determine population growth and evaluate the general health of the populations they study. These rates usually denote the number of births and deaths per 1,000 people in a given year.

16. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 16 Bruce Mayer, PE Chabot College Mathematics P2.3-56 Functions by Graphs  From the Graph the fcn T-table

17. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 17 Bruce Mayer, PE Chabot College Mathematics P2.3-56 Graph (f − g)(x)  Recall from Lecture (f − g)(x) = f(x) − g(x)  Next Use for Plotting y 1 = f(x) y 2 = g(x)  Thus for Plot f(x) − g(x) = y 1 − y 2

18. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 18 Bruce Mayer, PE Chabot College Mathematics P2.3-56 Graph (f − g)(x)  Recall from Lecture (f − g)(x) = f(x) − g(x)  Use the above relation to construct (f − g)(x) T-Table Can only Calc f(x) – g(x) where Domains OverLap

19. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 19 Bruce Mayer, PE Chabot College Mathematics P2.3-56 ( f – g )( x ) Graph

20. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 20 Bruce Mayer, PE Chabot College Mathematics All Done for Today Fcn Algebra By MicroProcessor

21. File = MTH55_Lec-04_ec_2-2_Fcn_Algebra.pp 21 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –