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DX Since functions convert the value of an input variable into the value of an output variable, it stands to reason that this output could then be used as an input to a second function. This process is known as composition of functions, in other words, combining the action or rules of two functions. FUNCTION COMPOSITION COMMON CORE ALGEBRA II Exercise #1: A circular garden with a radius of 15 feet is to be covered with topsoil at a cost of $1.25 per square foot of garden space. (a) Determine the area of this garden to the nearest square foot. Input = x In this exercise, we see that the output of an area function is used as the input to a cost function. This idea can be generalized to generic functions, fand g as shown in the diagram below. (d) (fog)(-2)= Output from f becomes the input to g (b) Using your answer from (a), calculate the cost of covering the garden with topsoil. There are two notations that are used to indicate composition of two functions. These will be introduced in the next few exercises, both with equations and graphs. Exercise #2: Given f(x)=x²-5 and g(x) = 2x+3, find values for each of the following. (a) f(g(1))= (b) g(ƒ(2)) = (c) g (g(0)) = (e) (gof)(3) = Final output=y (f) (ƒ•ƒ)(-1)=
Exercise #3: The graphs below are of the functions y=f(x) and y=g(x). Evaluate each of the following questions based on these two graphs. — y=f(=) (2) 8(ƒ(2)) = (d) (gof)(-2)= (b) f(g(-1))= (e) (fog)(0) - (1) x²-25 (2) x²-25 A Exercise #5: If f(x)=x² and g(x)=x-5 then f(g(x))- (3) x²-5 (4) x²-10x+25 - x= x(x) y On occasion, it is desirable to create a formula for the composition of two functions. We will see this facet of composition throughout the course as we study functions. The next two exercises illustrate the process of finding these equations with simple linear and quadratic functions. (b) g(f(x)) (c) g(g(1))= Exercise #4: Given the functions /(x)=3x-2 and g(x) = 5x+4, determine formulas in simplest y = ax+b form for: (1) (ƒ•ƒ)(0) - X CONOSCORE ALII UNIT 2-FUNCTIONS AS CORNERSTORTS OF ALGEBRAI-T43 CONDIOSCORE ALGIRII TIT? FUNCTION Ma Name: FLUENCY 1. Given /(x)=3x-4 and g(x)=-2x+7 evaluate: (a) f(g(0)) (b) g(f(-2)) (d) (gof)(6) FUNCTION COMPOSITION COMMON CORE ALGEBRA II HOMEWORK (d) A(A(0)) 2. Given h(x)=x+11 and g(x)=√x-2 evaluate: (a) h(g(18)) (b) g(h(4)) (e) (fog)(5) (a) h(k(-2)) (e) (hog)(38) (c) ƒ(ƒ(3)) (b) (kh)(0) (1) (gºg)(2) Date: (c) (g-g)(11) 3. The graphs of y=h(x) andy=k(x) are shown below. Evaluate the following based on these two graphs. y + (†) (g=h)(0) y=(1) (c) h(h(-2)) COMMOX COPE ALGERRA EL UNITAT-FUNCTIONS AS CORNERSTONES OF ALGERRI-Lessox (d) (kok)(-2)
4. If g(x)=3x-5 and h(x)=2x-4 then (goh)(x) = ? (1) 6x-17 (3) 5x–9 (2) 6x-14 (4) x-1 5. If f(x)=x² +5 and g(x)=x+4 then f(g(x)) = (1) x² +9 (3) 4x² + 20 (2) x² +8x+21 (4) x² +21 APPLICATIONS the function I (h)= 6. Scientists modeled the intensity of the sun, I, as a function of the number of hours since 6:00 a.m., h, using They then model the temperature of the soil, T, as a function of the intensity using the function 7(1)=√50001. Which of the following is closest to the temperature of the soil at 2:00 p.m. ? 12h-h² 36 (1) 54 (2) 84 (3) 67 (4) 38 7. Physics students are studying the effect of the temperature, T, on the speed of sound, S. They find that the speed of sound in meters per second is a function of the temperature in degrees Kelvin, K, by S(K)=√410K. The degrees Kelvin is a function of the temperature in Celsius given by K(C)=C+273.15. Find the speed of sound when the temperature is 30 degrees Celsius. Round to the nearest tenth. REASONING 8. Consider the functions f(x)=2x+9 and g(x)=*=9. Calculate the following. 2 (a) g (ƒ(15)) (b) g(f(-3)) (d) What appears to always be true when you compose these two functions? (c) g(f(x))