A function from a set D to a set R is a rule that assigns a unique element in R to each element in D.

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1 FUNCTIONS AND THEIR PROPERTIES Learning Targets 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function given an set of numbers, an equation, or a graph 3. Describe the tpe of discontinuit in a graph as removable or non-removable 4. For a given function, describe the intervals of increasing and decreasing 5. Label a graph as bounded above, bounded below, bounded, or unbounded. 6. Find all relative etrema on a graph 7. Understand the difference between absolute and relative etrema 8. Describe the smmetr of a graph as odd or even 9. Prove a function is odd, even, or neither 10. Given a rational equation, find vertical asmptotes and holes (if the eist) 11. Given a rational equation, find the end behavior model 12. Given a rational equation, describe the end behavior using end behavior and limit notation. 13. Given a rational equation, use the end behavior to find an horizontal asmptotes. 14. Given a rational equation, determine when slant (oblique) asmptotes eist and find them. This section is full of vocabular (obvious from the list above?). We will be investigating functions, and ou will need to answer questions to determine how each of these properties are applied to various functions. Function Definition and Notation In the last chapter, we used the phrase is a function of. But what is a function? In Algebra 1, we defined a function as a rule that assigned one and onl one (a unique) output for ever input. We called the input the domain and the output the range. Usuall, the set of possible -values is the domain, and the resulting set of possible -values is the range. Definition: Function A function from a set D to a set R is a rule that assigns a unique element in R to each element in D. To determine whether or not a graph is a function, ou can use the vertical line test. If an vertical line intersects a graph more than once, then that graph is NOT a function. Eample 1: A relation is ANY set of ordered pairs. State the domain and range of each relation, then tell whether or not the relation is a function. a) {( 3, 0),( 4, 2),( 2, 6)} b) {( 4, 2),( 4, 2),(9, 3) ( 9, 3)} Relation is a function. Relation is not a function. c) d) Relation is not a function. Does not pass a vertical line test. 1-1

2 For man functions, the domain is all real numbers, or. We tpicall start with this, and then see if there are an values of that cannot be used. The domain of a function can be restricted for 3 reasons that ou need to be aware of in this course: 1. NO negatives no negative numbers inside a square root 2. ** NO zeros no zeros in the denominator of a fraction 3. log NO negatives... and NO zeros. no negatives and no zeros inside a logarithm Eample 2: Without using a graphing calculator, what is the domain of each of the following functions? a) c) = ln , ,8 b) h ( ) = , 3 3, \3 + 2 d) = d Continuit In non technical terms, a function is continuous if ou can draw the function without ever lifting our pencil. Eample 3: Graph each of the following functions. What do ou notice? What happens when = 2 on the graph of b? a) f ( ) = + 2 b) g ( ) = *You can use Graphing Calculators Eample 4: What is the domain and range of the two functions above? 1-2

3 Eample 5: The following graphs demonstrate three tpes of discontinuous graphs. Label each tpe as removable or nonremovable. Informall we sa that f has a removable discontinuit if there is a hole in the function, but f has a non- removable discontinuit Asmptote- Non Removable Discontinuit Hole- Removable Discontinuit if there is a "jump" or a vertical asmptote. Jump - Non Removable Discontinuit Eample 6 : Tell where the function graphed below is discontinuous. Describe each discontinuit. Eample 7 : What is the domain and range of the function above? Eample 8: Using our calculator, graph each of the following functions. Determine if it has discontinuit at = 0. If there is a discontinuit, tell whether it is removable or non-removable. a) b) g ( ) = no discontinuit at =0 (an asmptote) Tpe equation here.there is non-removable discontinuit c) h ( ) = 3 d) k ( ) =

4 Increasing/Decreasing Functions While there are technical definitions for increasing and decreasing, just remember to read the graph LEFT to RIGHT. Eample 9: Using the graph below, what intervals is the function increasing? Decreasing? Constant? 0,1 1,2 constant 2,3 3,4 Boundedness You need to understand the difference between the following terms: BOUNDED BELOW: f is said to be bounded below if its range is bounded below. BOUNDED ABOVE: f is said to be bounded above if its range is bounded above. BOUNDED: f is said to be bounded if its range is bounded below and above. Local and Absolute Etrema Etrema is the plural form of one etreme value. Etrema is one word that includes maimums and minimums. Local Etrema (also called local) are points bigger (or smaller) than ever in some open interval. Absolute Etrema (also called global) are points bigger (or smaller) than ever in the domain we are working on. Eample: Suppose the following function is defined on the interval [a, b]. Identif the location and tpes of etrema. rel ma abs & rel ma rel min abs min a c d e b 1-4 Nothing happens here as relative etrema cannot occur on the end points of the domain.

5 Eample 10: Use our graphing calculator, to graph the function a) Identif all etrema. g ( ) = b) Identif the intervals on which the function is increasing, decreasing, or constant. Smmetr The net topic we concern ourselves with when dealing with functions is the idea of smmetr. Smmetr on a graph means the functions look the same on one side as it does on another. We are most concerned with the tpes of smmetr that can be eplored numericall and algebraicall in terms of ODD and EVEN functions. 3 Tpes of Smmetr Smmetr with respect to the -ais: EVEN FUNCTIONS Graphicall Numericall Algebraicall Graph it! = Plug a number Prove f(-)=f() Smmetr with respect to the origin: -6 ODD FUNCTIONS Graphicall Numericall Algebraicall 6 Graph it! = 3 Plug a number Prove f(-)=-f() Smmetr with respect to the -ais: NOT A FUNCTION Graphicall Numericall Algebraicall =±

6 Odd vs Even Functions While there are man functions out there that are neither even nor odd, our concern with odd and even functions is twofold #1: Identif graphs of functions that are Odd or Even #2: PROVE a function is Odd or Even While the first item above can be done graphicall, numericall, or algebraicall, the second is done ONLY algebraicall. Eample 11: Prove each function is odd, even, or neither. a) f ( ) = b) g ( ) = c) h ( ) = Vertical Asmptotes Eample 12: Graph the function f ( ) = There is a vertical asmptote. 1. What happens at = 1? Wh? 1 To describe a vertical asmptote, we must talk about what happens to the -values of a function as the -values get close to a certain number. We use the notation 1 to sa that is approaching 1. If we onl want to approach 1 from the right side, we use the notation If we onl want to approach 1 from the left side, we use the notation : In calculus, we use what is called limit notation lim f ( ) = ± or lim f ( ) = ±. c + c Vertical asmptotes occur in rational functions when denominator is 0. If ou plug a point into our function and get, ou should look for a removable discontinuit a.k.a. a hole. Horizontal Asmptotes and End Behavior Horizontal Asmptotes occur when the function values (-values) get close to a specific number as the -values get reall, reall large in the positive direction ( ) or negative direction ( ). As ±, we sa the end behavior of the function is a description of what f () approaches. IF f ( ) "a number" as ±, we sa the function has a horizontal asmptote. 1-6

7 Eample 13: Graph the function g ( ) = 4 3. Where does the horizontal asmptote occur? Eample 13: Graph the function 27. Where does the horizontal asmptote occur? 3 is a horizontal asmptote. The end behavior model of a polnomial is the leading coefficient and the highest power of the variable. A rational function is just two polnomials divided, so the end behavior model of a rational function is just the end behavior model of the numerator divided b the end behavior model of the denominator. 4 Eample 14: What is the end behavior model for g ( ) 3 =? How is this related to the horizontal asmptote. Eample 14: What is the end behavior model for 27. How is this related to the horizontal asmptote occur? Eample 15: Consider the function h ( ) = 1. As, where does h () approach? *PRECALCULUS STOP HERE CALCULUS CONTINUE Summar for Horizontal Asmptotes For rational functions we have the following results. If f ( ) = a m + L and g ( ) = b n + L then f ( ) takes on three different forms. g ( ) m = n m < n m > n End Behavior Model End Behavior Asmptote 0 0 Eample 16: Find the end behavior models of each function and determine what (if an) the horizontal asmptotes are

8 Calculus & Precalculus Eample 17: Go back and find the slanted (or oblique) asmptote for the graph in part b above. Question has been removed for now. Eample 18: Find ALL vertical and horizontal asmptotes for the following functions. a) f ( ) = c) h ( ) = (3 5) ( 8) b) g ( ) = d) k ( ) =