Relations: UNIT 5: FUNCTIONS STUDY SHEET Relation: A relation is simply a set o ordered pairs. The irst elements in the ordered pairs (the -values), orm the domain. The second elements in the ordered pairs (the y-values), orm the rane. Only the elements "used" by the relation constitute the rane. This mappin shows a relation rom set A into set B. This relation consists o the ordered pairs (1,2), (3,2), (5,7), and (9,8). The domain is the set {1, 3, 5, 9}. The rane is the set {2, 7, 8}. (Notice that 3, 5 and 6 are not part o the rane.) The rane is the dependent variable. The ollowin are eamples o relations. Notice that a vertical line may intersect a relation in more than one location. This set o 5 points is a relation. {(1,2), (2, 4), (3, 5), (2, 6), (1, -3)} Notice that vertical lines may intersect more than one point at a time. This parabola is also a relation. Notice that a vertical line can intersect this raph twice.
Functions: I we impose the ollowin rule on a relation, it becomes a unction. Function: A unction is a set o ordered pairs in which each -element has only ONE y-element associated with it. The relations shown above are NOT unctions because certain - elements are paired with more than one unique y-element. The irst relation shown above can be altered to become a unction by removin the ordered pairs where the -coordinate is repeated. It will not matter which "repeat" is removed. unction: {(1,2), (2,4), (3,5)} The raph at the riht shows that a vertical line now intersects only ONE point in our new unction. Vertical line test: each vertical line drawn throuh the raph will intersect a unction in only one location.
Functional Notation: Traditionally, unctions are reerred to by the letter name, but need not be the only letter used in unction names. The ollowin are but a ew o the notations that may be used to name a unction: (), (), h(a), A(t),... Note: The () notation can be thouht o as another way o representin the y-value in a unction, especially when raphin. The y-ais is even labeled as the () ais, when raphin. Evaluatin Functions: To evaluate a unction, simply replace (substitute) the unction's variable with the indicated number or epression. 1. A unction is represented by () = 2 + 5. Find (3). 2. To ind (3), replace the -value with 3. (3) = 2(3) + 5 = 11. The answer, 11, is called the imae o 3 under (). To ind (3h+2), replace the -values with 3h + 2. Usin parentheses or this substitution will help prevent alebraic errors. Use (3h + 2) when substitutin.
Domain and Rane: The domain is the set o all irst elements o ordered pairs (-coordinates). The rane is the set o all second elements o ordered pairs (y-coordinates). Domain and rane can be seen clearly rom a raph. Eample 1: Eample 2: Domain: {1, 3, 4, 6} Rane: {-2, 2, 5} Domain: Rane: (all real numbers)
Composition o Functions: The term "composition o unctions" (or "composite unction") reers to the combinin o unctions in a manner where the output rom one unction becomes the input or the net unction. In math terms, the rane (the y-value answers) o one unction becomes the domain (the -values) o the net unction. The notation used or composition is: and is read " composed with o " or " o o ". Notice how the letters stay in the same order in each epression or the composition. (()) clearly tells you to start with unction (innermost parentheses are done irst). Composition o unctions can be thouht o as a series o taicab rides or your values. The eample below shows unctions and workin toether to create the composition. Note: The startin domain or unction is bein limited to the our values 1, 2, 3 and 4 or this eample. Eamples: Now, suppose that we wish to write our composition as an alebraic epression. 1. Substitute the epression or unction (in this case 2) or () in the composition. This will clearly show you the order o the substitutions that will need to be made. 2. Now, substitute this epression (2) into unction in place o the -value. Perorm any needed simpliications (none needed in this eample). 1. Given the unctions and, ind a.) and b.) Answer: a.) b.) Notice that and do not necessarily yield the same answer. Composition o unctions is not commutative.
2. Given the unctions and, ind a.) and b.) Answer: a.) = h(p(3)) where p(3) ives an answer o 5 and h(5) then ives an answer o 25. The answer is 25. b.) One-to-One Function: A unction rom A to B is called one-to-one (or 1-1) i whenever (a) = (b) then a = b. No element o B is the imae o more than one element in A. In a one-to-one unction, iven any y there is only one that can be paired with the iven y. "One-to-One" NOT "One-to-One"
INVERSE FUNCTIONS: Basically speakin, the process o indin an inverse is simply the swappin o the and y coordinates. This newly ormed inverse will be a relation, but may not necessarily be a unction. Consider this subtle dierence in terminoloy: Deinition: INVERSE OF A FUNCTION: The relation ormed when the independent variable is echaned with the dependent variable in a iven relation. (This inverse may NOT be a unction.) Remember: The inverse o a unction may not always be a unction! The oriinal unction must be a one-to-one unction to uarantee that its inverse will also be a unction. Deinition: A unction is a one-to-one unction i and only i each second element corresponds to one and only one irst element. (each and y value is used only once) Use the horizontal line test to determine i a unction is a one-to-one unction. I ANY horizontal line intersects your oriinal unction in ONLY ONE location, your unction will be a one-to-one unction and its inverse will also be a unction. The unction y = 3 + 2, shown at the riht, IS a one-to-one unction and its inverse will also be a unction. (Remember that the vertical line test is used to show that a relation is a unction.) Deinition: The inverse o a unction is the set o ordered pairs obtained by interchanin the irst and second elements o each pair in the oriinal unction. Should the inverse o unction () also be a unction, this inverse unction is denoted by -1 (). Note: I the oriinal unction is a one-to-one unction, the inverse will be a unction. [The notation -1 () reers to "inverse unction". It does not alebraically mean 1/ ().] I a unction is composed with its inverse unction, the result is the startin value. Think o it as the unction and the inverse undoin one another when composed. Consider the simple unction () = {(1,2), (3,4), (5,6)} and its inverse -1 () = {(2,1), (4,3), (6,5)}
"So, how do we ind inverses?" Consider the ollowin methods: Swap ordered pairs: I your unction is deined as a list o ordered pairs, simply swap the and y values. Remember, the inverse relation will be a unction only i the oriinal unction is one-to-one. Eamples: a. Given unction, ind the inverse relation. Is the inverse relation also a unction? Answer: Function is a one-to-one unction since the and y values are used only once. Since unction is a oneto-one unction, the inverse relation is also a unction. Thereore, the inverse unction is: b. Determine the inverse o this unction. Is the inverse also a unction? 1-2 -1 0 2 3 4-3 () 2 0 3-1 1-2 5 1 Answer: Swap the and y variables to create the inverse relation. The inverse relation will be the set o ordered pairs: {(2,1), (0,-2), (3,-1), (-1,0), (1,2), (-2,3), (5,4),(1,-3)} Since unction was not a one-to-one unction (the y value o 1 was used twice), the inverse relation will NOT be a unction (because the value o 1 now ets mapped to two separate y values which is not possible or unctions).
Solve alebraically: Solvin or an inverse relation alebraically is a three step process: Eamples: a. Find the inverse o the unction Answer: Remember: Set = y. Swap the variables. Solve or y. 1. Set the unction = y 2. Swap the and y variables 3. Solve or y Use the inverse unction notation since () is a one-to-one unction. b. Find the inverse o the unction (iven that is not equal to 0). Answer: Remember: Set = y. Swap the variables. Eliminate the raction by multiplyin each side by y. Get the y's on one side o the equal sin by subtractin y rom each side. Isolate the y by actorin out the y. Solve or y. Use the inverse unction notation since () is a one-to-one unction.
Deinitions o operations o unctions: Sum Dierence Product Quotient 0 ) ( ; Composition ) (
Averae Rate o Chane: