Lecture Notes: Analyze Functions PDF

Chapter 1 Analyze Functions 1.1 Domain, Range and Endbehavior In this chapter, we want to introduce a formal definition of function. Definition 1.1.1. A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the codomain, such that each input i...

Chapter 1 Analyze Functions 1.1 Domain, Range and Endbehavior In this chapter, we want to introduce a formal definition of function. Definition 1.1.1. A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the codomain, such that each input is related to exactly one output. In mathematical terms, a function f from a set X (the domain) to a set Y (the codomain) is a rule that assigns to each element x in X exactly one element y in Y. This relationship is often denoted as f : X → Y and f (x) = y. That is to say, a function will satisfy the following conditions : 1. Domain : The set of all possible inputs for the function. 2. Codomain : The set of all possible outputs for the function, though not all elements of the codomain must be actual outputs. 3. Unique Association : Each element of the domain is associated with exactly one element of the codomain. Besides, there are some concepts similar to codomain : 4. Range : The set of all actual outputs for the function. 5. Image : The output corresponding to a specific input or a subset of inputs. That is, range of f = {y | ∀x ∈ X, and f (x) = y ∈ Y } and image of a subset A of X (or an element x of X), denoted by f (A) = {y | ∀x ∈ A, and f (x) = y ∈ Y } (or f (x)). Note : 8 CHAPTER 1. ANALYZE FUNCTIONS 9 In R, we sometime consider domain and range as an interval, we use the notation as : 1. Open interval : (a, b) := {x ∈ R | a < x < b} 2. Closed interval : [a, b] := {x ∈ R | a ≤ x ≤ b} 3. Not open or closed : [a, b) := {x ∈ R | a ≤ x < b} or (a, b] := {x ∈ R | a < x ≤ b} The end behavior of a function describes what happens to the values of f (x) as the x - values increase without bound (x → +∞), or as they decrease without bound (x → −∞). PS : In mathematics, ∞ is a symbol, not a number, so it is incorrect to say that x = ∞. We can only describe as x approaching infinity, denoted by x → ∞ (or x approaching negative infinity, denoted by x → −∞) Note : 1.2 Characteristics of Functions and Graphs In general, we are used to observing the behavior of a function through its graph. But we will leave the methods for drawing the graph until we introduce calculus. Now we want to introduce the behavior that we are concerned with. Definition 1.2.1. 1. Increasing : If for any x1 , x2 ∈ [a, b] with x1 < x2 , we have f (x1 ) ≤ f (x2 ), then we say f (x) is increasing on [a, b] (or f (x1 ) < f (x2 ), then f (x) is strictly increasing on pa, b]). 2. Decreasing : If for any x1 , x2 ∈ [a, b] with x1 < x2 , we have f (x1 ) ≥ f (x2 ), then we say f (x) is decreasing on [a, b] (or f (x1 ) > f (x2 ), then f (x) is strictly decreasing on [a, b]). 3. Turning point : It is a point where the function changes from increasing to decreasing or from decreasing to increasing. That is, if x0 ∈ [a, b] is a turning point, there exist a constant ε such that f (x) is increasing on [x0 , x0 + ε] and decreasing on [x0 − ε, x0 ] (or f (x) is decreasing on [x0 , x0 + ε] and increasing on [x0 − ε, x0 ]). 4. End point : If we say f (x) is define on [a, b], that is, the domain of f (x) is [a, b], then we will call a, b as the end points. Moreover, if f (x) is increasing / decreasing over its entire domain, the we call it as an increasing / a decreasing function. We can also refer to it as a monotonic function. Note :