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Advanced Functions and Equations Part I Composition and operations on functions, Transformations Sections 3.4, 3.5 Advanced Functions and Equations Composition and operations on functions Section 3.5 Operations of functions expl: A certain company has two factories. The first factory has a revenue of R1(x) = 3x + 40 where x is the number of units made. The second factory has a revenue of R2(x) = 4x + 50. 1.) What is the total revenue of both factories? 2.) How much more does factory 2 make? For part 1, you need to find the sum of the revenues. Algebraically, let’s say R1 ( x) R2 ( x) 3x 40 4 x 50 7 x 90 For part 2, you need to find the difference of the revenues. Algebraically, let’s say R2 ( x) R1 ( x) Put parentheses around 3x + 40 4 x 50 3x 40 because you’re subtracting. 4 x 50 3x 40 x 10 Notation The following notation is often used. ( f g )( x) f ( x) g ( x) ( f g )( x) f ( x) g ( x) ( f g )( x) f ( x) g ( x) f f ( x) ( x ) g , g ( x) 0 g ( x) Domains: all real numbers in both domains of f and g and, in the case of (f/g)(x), exclude those numbers that make g(x) = 0 expl: #5, pg 306 Let f ( x) x g ( x) 3x 5 Remember the domain will be those Find a.) f + g numbers that satisfy b.) f – g both functions. In the case of the c.) fg quotient, we exclude d.) f / g those x’s that would and their domains. cause division by zero. a.) f ( x) g ( x) b.) f ( x) g ( x) x 3x 5 x (3 x 5) x 3x 5 domain: x0 domain: x0 c.) f ( x) g ( x) d.) f ( x) x x 3x 5 g ( x) 3 x 5 5 domain: x 0, x domain: x0 3 Composition expl: A popular umbrella manufacturer has this information. P(x) = .0025x2 + .2x + 4, where P(x) represents the monthly profit of the company and x represents the number of umbrellas sold that month, x 0 U(t) = 80t + 20, where U(t) represents the number of umbrellas sold per month and t represents the number of days it rains that month, 0 t 31 We are interested in how profit is related to the number of days of rain. Let’s find the equation. A specific example Say we have had 12 days of rain this month. Find the number of umbrellas we can expect to sell and the profit this should yield. U (12) 80(12) 20 980 If it rains 12 days, we can expect to sell 980 …Now find P(980) umbrellas… the profit we make if we .0025(980) .2(980) 4 2 sell 980 umbrellas. .0025 960400 196 4 2601 If we sell 980 umbrellas, we will make a profit of $2601. Number of Number of umbrellas days of rain U(t) Number of Profit umbrellas P(x) We’d like to bypass the number of umbrellas stage. We’d like a formula that gives us profit given number of days of rain. P( x) .0025x 2 .2 x 4 …where x represents the number of umbrellas. But U(t) also represents the number of umbrellas. So we can substitute U(t) in for x to find P(U(t)). Apply the rule of P(U (t )) P(x) to the number P(80t 20) 80t + 20 .002580t 20 .280t 20 4 2 .0025 6400t 2 3200t 400 .280t 20 4 16t 2 8t 1 16t 4 4 16t 2 24t 9 So P (t ) 16 t 24 t 9 where t is the number of 2 days of rain and P(t) is the profit. We used composition of functions to find P(U(t)). Now we know the relationship between profit and number of days of rain. Check the specific example. If it rains 12 days, what’s the profit? P(12 ) 16 (12 ) 24 (12 ) 9 2601 2 Composition notation: ( f g )(x) f ( g ( x)) expl: #14 ac, pg 306 Let f ( x) 3x 2 Remember f(x) and g ( x) 2 x 1 2 g(x) are rules. They tell you what to do a.) Find f g 4 . to x, no matter c.) Find f f 1 . what that value is called. Verify results (will do part a on grapher). For part a, f g 4 f ( g ( 4)) g(4) = 2(4)2 – 1 = 31 f (31) 3(31) 2 For part c, 95 f f 1 f ( f (1)) f(1) = 3(1) + 2 = f (5) 5 3(5) 2 17 Verify part a on grapher f g x First, find the f ( g ( x )) equation for y f g x f (2 x 1) 2 3( 2 x 1) 2 2 6x 3 2 2 6x2 1 Then graph it and use the calculator to calculate the y value when x is 4. On the Graph screen, press 2nd TRACE (CALC) to get to the Calculate menu. Then select Value. Give it an input of 4 for x, it will tell you what y is. In order for the Value or EVAL on the EVAL function to work, your TI-85 or 86 x value must be within the window dimensions. Domain of f ( g ( x)) all real numbers x that are in the domain of g and such that g(x) is in the domain of f. expl: Find f(g(x)) and determine its domain. f ( x) x g ( x) 3 x 2 First find f(g(x)). f ( g ( x)) f (3x 2) 3x 2 The domain of this composite function are those numbers that would work in g(x) = 3x + 2, all real numbers, and those g(x) values that would work in f. Now f requires that its inputs are not negative. So… Domain: 3x 2 0 2 x 3 You could also think of this as simply finding the domain, those numbers that work, of the 3x y function2 Look at its graph! 2 Domain: x 3 expl: # 52 Show that f(g(x)) = x and g(f(x)) = x for f ( x) 4 3 x g ( x) 4 x 1 We will see later how this 3 proves the two functions are inverses. Show f(g(x)) = x. Show g(f(x)) = x. Apply the f ( g ( x)) function’s g ( f ( x)) 1 rule g (4 3x) f 4 x 3 4 ( 4 3 x) 1 1 3 4 3 4 x 4 4 3 x 3 1 4 4 x 3 Simplify. 3 x 1 44 x Watch the order of 3 x operations! x Worksheet “Functions: Composition and operations” provides practice in performing operations and compositions as well as determining domain. 3.5 homework: 1, 9, 13, 27, 32, 33, 47, 51, 55, 65, 67, 77 Advanced Functions and Equations Transformations Section 3.4 Worksheets “Transformations” introduces several transformations. Most of the functions we will deal with this semester are transformations of the functions discussed in “Library of functions”. These functions, such as y = x 2, y = x, and others, are called mother functions. “Part II” of this worksheet summarizes the transformations for which you will be responsible. “Transformations 2” and “Library of functions and transformations 2” provide more practice. Intro to Transformations worksheet Part I: Graph carefully. Check your values with your neighbors often to make sure you are on the right track. 1. Complete the table and graph f ( x) x and h( x) x 2 3 . 2 x -3 -2 -1 0 1 2 3 f ( x) x 2 h( x ) x 3 2 x -3 -2 -1 0 1 2 3 f ( x) x 2 9 4 1 0 1 4 9 h( x) x 2 3 12 7 4 3 4 7 12 In the table, each h(x) Notice how value is 3 for each x, more than the the point is correspondin moved up g f(x) value 3 units to because we make h(x). added 3 to each f(x). Transformations worksheet Let c be a positive Transformation of Number of question real number. graph which is example c f ( x), c 1 Vertical stretch by 5 factor of c Vert. compression c f ( x), 0 c 1 by factor of c 6 Horizontal shift f ( x c) to right c units 4 Horizontal shift f ( x c) to left c units 3 Transformations worksheet cont. Let c be a positive Transformation of Number of question real number. graph which is example f ( x) c Vertical shift up c units 1 f ( x) c Vertical shift 2 down c units f (x) Reflection about 7 x axis f ( x) Reflection about 8 y axis You can think through transformations by imagining what is happening to the y values. For instance, if you compare y1 x and y 2 5x , you’ll notice the second 2 2 function’s y values are merely five times the first function’s y values. So each point is stretched away from the x-axis. The graph will be elongated. So we say the graph is vertically stretched by a factor of 5. Check it out by graphing both on your calculator! 3.4 homework: 1 – 12, 13 – 16, 17, 19, 21, 23, 27, 33, 46, 53, 63, 69

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