# OSU MTH 111 - Act-#13 Composition-Key (5 pages)

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## Act-#13 Composition-Key

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- School:
- Oregon State University
- Course:
- Mth 111 - College Algebra

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Composition of Functions Math 111 Name GrpAct 13 Sect 12 4 16 Prerequisite Skills Evaluating functions Determining Domain Range Graphing a function point wise Key Terms Composition of functions Inside function Outside function Learning Objectives Use composition of functions to model multistep processes Compose functions defined numerically graphically and symbolically Interpret the meaning of the output of composed functions Determine the domain of a composition of functions 1 In the United States temperature is commonly measured in degrees Fahrenheit However in most sciences temperature 5 F 32 accepts an input of temperature in degrees Fahrenheit 9 and outputs the corresponding temperature in degrees Celsius The function K C C 273 accepts a temperature is measured in Celsius or Kelvin The function C F measured in Celsius and outputs the corresponding temperature measured in Kelvin a Use the formulas above to complete the following temperature conversion tables Fahrenheit Celsius Celsius Kelvin 13 25 25 248 5 15 10 263 32 0 0 273 41 5 15 288 59 15 25 298 77 25 40 313 b Fill out the table below converting temperatures directly from Fahrenheit to Kelvin Fahrenheit Kelvin 13 248 32 273 41 278 77 298 104 313 122 323 c Give a formula for a function that will accept an input of temperature in degrees Fahrenheit and will output the corresponding temperature in Kelvin The function you created in part c is the composition of the functions C and K and is written as 5 5 2297 K C F C F 273 F 32 273 F 9 9 9 This is read K composed with C of F or K of C of F In a composition the output of a function becomes the input of another function see the diagram in the box below d Use the formula you found in part c to determine K C 50 and explain what this represents Include units 5 2297 250 2297 2547 50 283 K 9 9 9 9 5 5 K C 50 50 32 273 18 273 10 273 283 K 9 9 K C 50 K C 50 OR A Composition of Functions The diagram below illustrates the composition K C F K C F original input fnt output input ftn final output C F F C C F K K Temp in Fahrenheit Temp in Celsius Check Your Work You should get K C 50 283 Make sure to fully explain what this represents Temp in Kelvin 2 The functions f and g are graphed below The function f accepts as input a UV index and outputs the time it takes for the skin of an average person to begin showing damage from sunburn The function g accepts as input the time of day measured as hours after 6 am and outputs the UV index for a typical spring day in Waikiki HI f 7 12 a Find and interpret f 7 Include units This means that when the UV index is 7 it only takes 12 minutes until skin damage begins g 10 4 b Find and interpret g 10 This means that 10 hours past 6 AM at 4 O clock the UV index is 4 UV units c If a person starts sunbathing at 2 pm in Waikiki how long will it take to begin showing sunburn damage About 7 minutes d Sketch a diagram like the one in the box for problem 1 d illustrating the composition of functions f g x t g g t f g f t Hours since 6am UV index units Time until skin damage in minutes e Find and interpret f g x Include units f g 10 f g 10 f 4 17 This means that at 10 hrs past 6 AM at 4pm the sun will begin to damage your skin in 17 minutes f On the axes at right sketch a graph of f g x Place appropriate labels including units on each axis Use table below x 1 3 5 6 7 9 11 f g x 45 17 7 5 7 17 45 g Explain why the shape of the graph makes sense in terms of the context Because moving forward from 6 am the amount of time you can spend in the sun before damage gets shorter and shorter until 6 hrs past 6 am noon where it is 5 minutes and then it starts to get longer and longer 3 The tables below define two functions f and g x f x 3 1 5 3 2 0 4 1 x g x 4 2 0 3 3 5 6 1 a Compute each of the following or explain why it is undefined This diagram illustrates a composition of functions f g 3 f 1 0 g f 3 g 2 5 g f 2 Thinking about the domain of a composition of functions g f 5 g 4 g f 2 g Undefined Undefined g f 1 g 0 6 f g 4 f g 4 f 3 1 If x is not in the domain of f then f x does not exist and g f x does not exist So if x is not in the domain of f it is also not in the domain of NOTE for parts b and c The functions f and g are defined by their table b What is the domain and range of f D f 3 1 5 3 Similarly if f x is not in the domain of g then g f x does not exist and x is not in the domain of R f 2 0 4 1 c What is the domain and range of g Dg 4 2 0 3 Rg 3 5 6 1 Rg f 5 6 d What is the domain of g f x Need help Sketch a diagram to illustrate this composition Dg f 3 1 4 Let f x 1 and g x 2 x 9 x2 a What is the domain of f 0 0 b What is the range of f 0 c What is the domain of g 9 d What is the range of g 0 e Give a formula for f g x f g x f What is the domain of f g When computing a formula for a composition it is often easier to see what the domain of that composition will be if we don t simplify For instance on part e of 5 determine the domain part f BEFORE you simplify the formula for the composition in part e This should make it easier to answer part f 1 4x 36 D f g 9 g Explain how you found the domain of f g in part f First I found the domain of the inside function g then I looked to see if all of the outputs of g could bre used as inputs for f The output of 0 from g cannot be put in f so I need to exclude the input to g that gives g an output of 0 which is 9

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