CHapter1 Section 2 Functions and their properties

Objectives Students will learn about Function definition and notation Domain and range Continuity Increase and decreasing functions Boundedness Local and absolute extrema Symmetry End behavior

A bit of history What is the word function ? Where does it comes from? The word function in the mathematical sense is attributed to Gottfried Leibniz ( ). One of the pioneers in the methods of Calculus. His intentions to clarify of notation is one his greatest contributions to scientific progress. Which is why we still his notations in calculus courses today. Ironically, it was not Leibniz but Leonhard Euler that introduce the familiar notation f(x).

Definition of function There are many definitions of functions but the one that we are going to study is the following The definition of a function from a set D to a set R is a rule that assigns to every element of D a unique element of R Question : What does that mean?

Examples Formal Definition of a Function A function relates each element of a set with exactly one element of another set (possibly the same set). A Function is Special But a function has special rules: It must work for every possible input value And it has only one relationship for each input value This can be said in one definition:

functions 1.1. "...each element..." means that every element in X is related to some element in Y. We say that the function covers X (relates every element of it). (But some elements of Y might not be related to at all, which is fine.) 2."...exactly one..." means that a function is single valued. It will not give back 2 or more results for the same input. So "f(2) = 7 or 9" is not right! The Two Important Things!

functions Note: "One-to-many" is not allowed, but "many-to-one" is allowed: (one-to-many) (many-to-one) This is NOT OK in a function But this is OK in a function When a relationship does not follow those two rules then it is not a function... it is still a relationship, just not a function.

What you think? If the following relationship a function Example: The relationship x → x 2

solution X: xY: x Could also be written as a table: It is a function, because: Every element in X is related to Y No element in X has two or more relationships So it follows the rules. (Notice how both 4 and -4 relate to 16, which is allowed.)

What do you think? Example: This relationship is a function

Solution It is a relationship, but it is not a function, for these reasons: Value "3" in X has no relation in Y Value "4" in X has no relation in Y Value "5" is related to more than one value in Y (But the fact that "6" in Y has no relationship does not matter)

Domain The set D of all input values is the domain of the function

Range The set R of all output values is the range of the function

In this illustration: the set "A" is the Domain, the set "B" is the Codomain, and the set of elements that get pointed to in B (the actual values produced by the function) are the Range, also called the Image. There are special name for what can go into, and what can come out of a function: What can go into a function is called the Domain What may possibly come out of a function is called the Codomain What actually comes out of a function is called the Range

solution In that example: Domain: {1, 2, 3, 4} Codomain: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Range: {3, 5, 7, 9}

Vertical line test Vertical Line Test On a graph, the idea of single valued means that no vertical line ever crosses more than one value. If it crosses more than once it is still a valid curve, but is not a function.

Example

Example

Continuous A function is continuous when its graph is a single unbroken curve that you could draw without lifting your pen from the paper. That is not a formal definition, but it helps you understand the idea. Here is a continuous function:

So what is not continuous (also called discontinuous) ? Look out for holes, jumps or vertical asymptotes (where the function heads up/down towards infinity). Not continuous( vertical asymptote)

Not Continuous (hole) (jump)

calculus A function f is continuous at x=q. if lim f(x)=f(a)

Increasing and decreasing A function f is increasing if, for any two points in the interval, a positive change in x results in a positive change in f(x). A function f is decreasing on an interval if, for any two points in the interval, a positive change in x results in a negative change in f(x) A function f is constant on an interval if, for any two points in the interval, a positive change in x results in a zero change in f(x)

A function is "increasing" when the y-value increases as the x-value increases,function like this: Decreasing Functions The y-value decreases as the x-value increases:

Example An Example Let us try to find where a function is increasing or decreasing. Example: f(x) = x 3 -4x, for x in the interval [-1,2]

Example

bounded A function f is bounded below if there is some number b that is less than or equal to every number in the range – lower bound A function f is bounded above if there is some number b that is greater than or equal to every number in the range- upper bound A function f is bounded if it is bounded both above and below

Absolute and local extrema A local maximum of a function f is a value f(c) that is greater than or equal to all range values of f on some open interval containing c. A local minimum of a function f is a value f(c) that is less than or equal to all range values of f on some open interval containing c.

Example

Even and odd functions They got called "even" functions because the functions x 2, x 4, x 6, x 8, etc behave like that, but there are other functions that behave like that too, such as cos(x): function is "even" when: f(x) = f(−x) for all x In other words there is symmetry about the y-axis (like a reflection):symmetry about the y-axis This is the curve f(x) = x 2 +1 Cosine function: f(x) = cos(x) It is an even function

Even functions But an even exponent does not always make an even function, for example (x+1) 2 is not an even function.

Odd functions A function is "odd" when: −f(x) = f(−x) for all x Note the minus in front of f: −f(x). And we get origin symmetry:origin symmetry This is the curve f(x) = x 3 −x

They got called "odd" because the functions x, x 3, x 5, x 7, etc behave like that, but there are other functions that behave like that, too, such as sin(x): Sine function: f(x) = sin(x) It is an odd function But an odd exponent does not always make an odd function, for example x 3 +1 is not an odd function.

Neither Odd nor Even Don't be misled by the names "odd" and "even"... they are just names... and a function does not have to be even or odd. In fact most functions are neither odd nor even. For example, just adding 1 to the curve above gets this: This is the curve f(x) = x 3 −x+1 It is not an odd function, and it is not an even function either. It is neither odd nor even!

Example Example: is f(x) = x/(x 2 −1) Even or Odd or neither? Let's see what happens when we substitute −x: So f(−x) = −f(x) and hence it is an Odd Function

Example

Asymptote A line that a curve approaches, as it heads towards infinity: Types There are three types: horizontal, vertical and oblique:

It is a Horizontal Asymptote when: as x goes to infinity (or −infinity) the curve approaches some constant value b Horizontal Asymptotes It is a Vertical Asymptote when: as x approaches some constant value c (from the left or right) then the curve goes towards infinity (or −infinity). It is an Oblique Asymptote when: as x goes to infinity (or −infinity) then the curve goes towards a line y=mx+b (note: m is not zero as that is a Horizontal Asymptote). Vertical Asymptotes Oblique Asymptotes

Example Example: find asymptotes (x 2 -3x)/(2x-2)

Example

Summary Graphs are continuous and smooth Even exponents behave the same: above (or equal to) 0; go through (0,0), (1,1) and (−1,1); larger values of n flatten out near 0, and rise more sharply. Odd exponents behave the same: go from negative x and y to positive x and y; go through (0,0), (1,1) and (−1,−1); larger values of n flatten out near 0, and fall/rise more sharply Factors: Larger values squash the curve (inwards to y-axis) Smaller values expand it (away from y-axis) And negative values flip it upside down Turning points: there are "Degree − 1" or less. End Behavior: use the term with the largest exponent

Homework Do odd numbers from 1-6, 9-19,47-51, 55-58