Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics §1.1 Intro toFunctions Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

§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

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 

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

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

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}

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)}.

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.

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.

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.

Example  Is Relation a Fcn?

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)}.

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.

Example  Is Relation a Fcn?

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, 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)}.

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 uniquey 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.

Function Forms • Functions can be described by: • A Table • A Graph

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

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

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

Example  is an EQN a FCN?? • Determine whether each equation determines yas a function of x. a. 6x2 – 3y = 12 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

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 ycorrespond to the same value of x so the expression does NOT define y as a function of x.

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 allx’s that produce VALID Functional RESULTS

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}

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, ∞)

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, ∞)

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 (−∞, ∞)

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

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

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:

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”).

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 off○gconsists of those values x in the domain of g for which g(x) is in the domain of f

COMPOSITION OF FUNCTIONS • Graphically thef○gDomain Chain

COMPOSITION OF FUNCTIONS • Conceptually the f○g Operation Chain

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

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

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

Example  Fcn Composition • Given f(x) = 4x and g(x) = x2 + 2, find • SOLUTION = g(4x) = (4x)2 + 2 = 16x2 + 2 • This example shows that in general

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

Example  Fcn Composition • Given: • Solution b.

Example  Fcn Composition • Given: • Solution c.

Example  Composite Domain • Given: • Solution a.

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

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

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

Decomposing a Function • Solution:

WhiteBoard Work • Problems From §1.1 Exercise Set • 37, 65

All Done for Today SomeStatinDrugs

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege