7 Persamaan, Fungsi Eksponen, dan Logaritma



Worksheet 2: Composition of functionsName:Date:Let there be two functions defined as: f:A→B by f(x) for all x∈A g: B→C by g(x) for all x∈B Then, the new function, “gof” read as "g circle f" or "g composed with f", is defined as:, for all x∈A Range of g(x)Range of h(x) = g(f(x))Range of f(x)Domain of g(f(x))Domain of f(x)Domain of h(x)Example 1:Let two functions be defined as: f={(1,2) ,(2,3) ,(3,4) ,(4,5) } and g={(2,4) ,(3,2) ,(4,3) ,(5,1) } Check whether “gof” and “fog” exist for the given functions.Solution:Domain Range Hence, ?Range of “f”?Domain of “g” “gof”exists.?Range of “g”?Domain of “f” “fog” exists.f{1,2,3,4}{2,3,4,5}g{2,3,4,5}{4,2,3,1}={1,2,3,4}It means that both compositions “gof” and “fog” exist for the given sets.Example 2:Given and, find:( f o g)(x). ( g o f)(x). ( f o f)(x). ( g o g)(x). Solution:. Note that:That is, ( f o g)(x) is not the same as (g o f )(x). The open dot "o" is not the same as a multiplication dot "?", nor does it mean the same thing. f(x) ? g(x) = g(x) ? f(x) [always true for multiplication]...you cannot say that:( f o g)(x) = (g o f )(x) [generally false for composition]Domain and range of the composition of functionsConsider the function:when Domain of is , i.e. all real numbers but 1.Let us now see the expression of composition of function with itself, valid for real values of x≠0. Since f is undefined for x = 1, and is undefined for x = 0, thus the domai n of the composition is : ; i.e. all real numbers except 0 and 1. Sometimes you have to be careful with the domain and range of the composite function. General rule to determine the domain:PolynomialDomain , for ,,for Example:Given and , find the domains of ( f o g)(x) and (g o f )(x). Solution: So: Hence, the domain of ( f o g)(x) is "all x > 3".Now do the other composition: Hence, the domain of (gof)(x) is …Going backward: given composed function, find original functions Usually composition is used to combine two functions. But sometimes you are asked to go backwards. That is, they will give you a function, and they'll ask you to come up with the two original functions that they composed. Example 1:Given , determine two functions f (x) and g(x) which, when composed, generate h(x). Solution:This is asking you to notice patterns and to figure out what is "inside" something else. In this case, this looks similar to the quadratic, except that, instead of squaring x, they're squaring x + 5. So let's make g(x) = x + 5, and then plug this function into:Then h(x) may be stated as the composition of and g(x) = x + 5.Example 2:Given , determine two functions f (x) and g(x) which, when composed, generate h(x). Solution:Since the square root is "on" (or "around") the "3x + 4", then the 3x + 4 is put inside the square root, that is:Thus, g(x) = 3x + 4, , and h(x) = ( f o g)(x).ExerciseFor the given functions:f(x) = x + 1 , g(x) = 3x f(x) = 2x + 1 , g(x) = x2Find:Domain and range of each f(x) and g(x)Domain = Range = Domain = Range = Domain = Range = Domain = Range = Domain = Range = Domain = Range = Domain = Range = Domain = Range = Determine and its domain Domain = Domain = Domain = Domain = Determine and its domain Domain = Domain = Domain = Domain = Determine and its domain Domain = Domain = Domain = Domain = Determine and its domain Domain = Domain = Domain = Domain = A function is defined for real values by : for all real values except x =1 . Determine and draw the graph of resulting composition!Given f(2) = 3, g(3) = 2, f(3) = 4 and g(2) = 5, evaluate (f o g)(3)!Functions f and g are as sets of ordered pairs f = {(-2,1),(0,3),(4,5)} and g = {(1,1),(3,3),(7,9)} Find the composite function defined by g o f and describe its domain and range. Write function F given below as the composition of two functions f and g, where and Evaluate f(g(h(1))), if possible, given that and .For the composite function and , find !, , , For the composite function and , find !, , , ................
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