1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§1.1 Intro to Functions
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. §1.1 Learning Goals
Identify the domain of a function, and evaluate a function from an equation
Gain familiarity with piecewise-defined functions
Introduce and illustrate functions used in economics
Form and use composite functions in applied problems

3. ReCall the Ordered-Pair
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; e.g.,
An ordered pair that satisfies an equation is called a solution of the eqn 

4. 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. Mathematical RELATION
Any SET of ordered pairs is called a relation.
The set of all first components is called the domain of the relation,
The set of all SECOND components is called the RANGE of the relation

6. 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 (Domain)
The DOMAIN is the set of all first components, or {Titanic, Star Wars IV, Shrek 2, E.T., Star Wars I, Spider-Man}

7. 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 (Range)
The RANGE is the set of all second components, or {$600.8, $461.0, $441.2, $435.1, $431.1, $403.7)}.

8. FUNCTION Defined
A function is a “Rule” which “takes” a set X to a set Y, and is a relation in which each element of X corresponds to ONE, and ONLY ONE, element of Y. X Y X Y

9. 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. 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).
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. Example  Is Relation a Fcn?
IS a Function as each launch-point has EXACTLY ONE target

12. Example  Is Relation a Fcn?
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. Example  Is Relation a Fcn?
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. Example  Is Relation a Fcn?

15. Example  Is Relation a Fcn?
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, ), (Malcolm, ), (MARIA, ), (MARIA, ), (MARIA, ), (Ellen, ), (Duane, )}.

16. 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. Graph Created by Plotting the Points in the Table
Function Forms
Functions can be described by:
A Table
A Graph
Graph Created by Plotting the Points in the Table

18. Function Forms Math Functions are MOST OFTEN described by: An EQUATION
The Eqn can be used to MAKE a Table or Graph
NOTE: f(x) ≠ “f times x”
f(x) indicates EVALUATION of the function AT the INDEPENDENT variable-value of x

19. Evaluating a Function
Let g be the function defined by the equation  y = g(x) = x2 – 6x + 8
Evaluate each function value:
SOLUTION

20. Evaluating a Function
Evaluate fcn  y = g(x) = x2 – 6x + 8
SOLUTION

21. Evaluating a Function
Evaluate fcn  y = g(x) = x2 – 6x + 8
SOLUTION

22. Example  is an EQN a FCN??
Determine whether each equation determines y as a function of x.
a. 6x2 – 3y = b. y2 – x2 = 4
SOLUTION a.
any value of x corresponds to ONE value of y so it DOES define y as a function of x
Isolate y and analyze result

23. Example  is an EQN a FCN??
Determine whether each equation determines y as a function of x.
a. 6x2 – 3y = 12 b. y2 – x2 = 4
SOLUTION b.
TWO values of y correspond to the same value of x so the expression does NOT define y as a function of x.
Plan: Isolate y, then examine result for one-x corresponding to multiple y’s

24. 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. Example  Find the Domain
Find the DOMAIN of each function.
SOLUTION
f is not defined when the denominator is 0.
1−x2 ≠ 0 → Domain: {x|x ≠ −1 and x ≠ 1}

26. 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. 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. 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. Composite Functions
In the real world, functions frequently occur in which some quantity depends on a variable that, in turn, depends on yet another variable.
Functions such as these are called COMPOSITE FUNCTIONS

30. Composing a Function Composition with sets A & B by fcns g & f1 3 7 4 10 22 −1 2 8 h
h(x) = ?

31. Composing a Function
From The Diagram notice that since f takes the output from g we can combine f and g to get a function h:
f (g (x)) = f (3x + 1)
This Yields an eqn for h:

32. Composing a Function
The function h is the composition of f and g and is denoted f○g (read “the composition of f and g”, or “f composed with g”, or “f circle g”).

33. COMPOSITION OF FUNCTIONS
If f and g are two functions, the composition of function f with function g is written as f○g and is defined by the equation
The function where the domain of f○g consists of those values x in the domain of g for which g(x) is in the domain of f
ORDER is important in f○g

34. Fcn Comp → NOT Commutative
ORDER is important in this expression
In General:
𝑓∘𝑔 𝑥 =𝑓 𝑔 𝑥 ≠ 𝑔∘𝑓 𝑥 =𝑔 𝑓 𝑥
From Before: 𝑓∘𝑔 𝑥 = 3 2 𝑥− 5 2
But 𝑔∘𝑓 𝑥 = 𝑥−3 +1= 3 2 𝑥−8

35. COMPOSITION OF FUNCTIONS
Graphically the f○g Domain Chain

36. COMPOSITION OF FUNCTIONS
Conceptually the f○g Operation Chain

37. Example  Evaluate Composites
Given:
Find Each of the Following
Solution a.

38. Example  Evaluate Composites
Solution b.
Solution c.
Solution d.

39. Example  Fcn Composition
Given f(x) = 4x and g(x) = x2 + 2, find
SOLUTION
= f (x2 + 2)
= 4(x2 + 2)
= 4x2 + 8

40. Example  Fcn Composition
Given f(x) = 4x and g(x) = x2 + 2, find
SOLUTION
= g(4x)
= (4x)2 + 2
= 16x2 + 2
These examples confirm that in general

41. Example  Fcn Composition
Given:
Find Each Composite Function
Solution a.

42. Example  Fcn Composition
Given:
Solution b.
as 2𝑥+1 2 =4 𝑥 2 +4𝑥+1

43. Example  Fcn Composition
Given:
Solution c.

44. Example  Composite Domain
Given:
Solution a.

45. Example  Composite Domain
Given:
Solution b.
Soln c.
Domain: (−∞, 0)U(0, ∞) or {x|x ≠ 0}

46. Example  Composite Domain
Given:
Soln d.
Domain: (−∞, −1)U(−1, ∞) or {x|x ≠ −1}

47. DEcomposing a Function
Given:
Show that each of the following provides a DEcomposition of H(x)

48. DEcomposing a Function
Solution:

49. DEcomposing a Function
Solution:

50. WhiteBoard Work
Problems From §1.1 Exercise Set
37, 65

51. Some Statin Drugs All Done for Today

52. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –

54. P1-37

58. Function Equality Two functions f and g are equal if and only if:
f and g have the same domain
and
f(x) = g(x) for all x in the domain.