Functions, function notation and (IGCSE composite functions)

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1 New concepts and terms: (see below for explanations) Function Function notation Composite function f(g(x)) or f o g(x) Functions What is a function? A function is a relationship for which every input has onlyone unique output. Ex.A The relation G has the rule: the output is a multiple of the input Input x: Output y: G G is not a function because elements from the input have more than one output. Ex. is in relation with elements (,) (,) (,8) also, is in relation with elements (,), (,8) Ex.B The relation S has the rule: the output is double the input Input x: Output y: 5 S 8 10 S is a function because each element in the input is in relation with only one element in the output Ex.C Is the relation P a function? Input x: Output y: P 8 1
2 } } h1 Function notation Functions are generally represented by the letters f, g, h, The output y of an element from the input, x, is written f(x). or as f : x a The point (x, y) can therefore be written as (x, f(x)) Ex. The function f has the rule the output is triple the input f 1  The equation of the function is written: y = x or f ( x) = x or f : x a x We say: f of x is equal to x Or in the function f, x is mapped onto x Notice that f ( 0) = 0, f ( 1) =, f ( ) =, etc. f ( ) = input (x) output (y) On the Cartesian plane: Ex. D. Decide if each of the following graphs represents a function or not. Why or why not?
3 Cartesian graph of a function Given the graph of a relation, this relation is a function if the shadow of a vertical line crosses the graph at most at one point at a time. (If the shadow of a vertical line ever crosses the graph more than once at any one time, the graph is not a function) This is known as the vertical line test or v.l.t.
4 Graphing functions To graph a function, (or any relationship) fill out the table of values. Let x be any reasonable input, and find the corresponding y coordinate by calculating the function rule. Ex.E. Given the following function rules (Funktionen Gleichungen),fill in the tables of values and graph: a. g( x) = x b. h : x a x c. f ( x) = 1 x x y x y x y
5 Composite Functions Composite functions: f(g(x)) means that the entire result of g(x) is the input for f g(f(x)) means that the entire result of f(x) is the input for g Ex. F. Given: f ( x) = x + 1 g( x) = x h( x) = x, find: i. f(g()) ii. f(g(x)) iii. g(h(x)) iv. h(f(5)) v. g(f(x1)) vi. g(h(x)) 5
6 Practice Questions 1. Indicate if each of the following relations is a function. If not, justify your answer.. Indicate if each of the following relations is a function. If not, justify your answer.. Indicate which of the following graphs (R1, R, etc) represent functions:. Graph the following functions: a. g( x) = x b. h : x a x c. f ( x) = 1 x d. m( x) = x e. p : x a x 1 f. q( x) = 1 x + 5. Given: f ( x) = x + g( x) = x + 1 h( x) = x, find: a. f(g()) b. f(g()) c. g(f(0)) d. g(h()) e. f(g(x)) f. g(f(x)) g. f(h(x)) h. f(x+1) i. g(f(x+1))
7 . (LS p Q) Handelt es sich bei der Zuordnung um eine Funktion? Begründe deine Antwort. Skizziere im Falle einer Funktion einen Funktionsgraphen. a. Gefährene Strecke a Benzinverbrauch b. Benzinverbrauch a gegahrene Strecke c. Zeit a Körpergröße eines Menschen d. Körpergröße eines Menschen a Zeit e. Quadrat einer rationalen Zahl a Zahl f. Rationale Zahl a Quadrat der Zah (LS. P0 Q) 9. (LS p 0 Q) Welche Funktion gehört zur Wertetabelle? Gib eine Funktionsgleichung an. Ergänze gegebenenfalls in deinem Heft die fehlenden Werte in der Tabelle. 7
8 Answers 1a. Function, b. function, c. not a function (b leads to 1 and also to ) a. function, b. not a function (a leas to 1 and ), c. function, d. not a function ( a leads to 1, and ) a. function, b. function, c. not a function, d. not a function, e. function, f. not a function a. g q m p f h 5a. 1 b.  c. 0 d. e. x8 f. x+0 g. ¾ x+ h. x+ i. x+18 8
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