CE 1010 HW: S13 C siti n F ncti ns
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1 Name CE 1010 HW: S13 C siti n F ncti ns 1. Functions f and g are defined in the tables below. x f(x) x g(x) Find the values for these composition functions. a. f g( ) ( 9 ) b. g f ( ) ( 1 ) c. f g( ) ( 0 ) 2. Given f ( x) = 3x 1 and gx ( ) = x+ 5, find a. f( g( t3 )) b. g f ( ) ( 10 ) c. f( g( x) ) d. g( f( x)) 3. Simplify each function by applying the properties of exponents. a. y 4 5 = ( x ) b. y 3 2 = ( 2 x ) c. y = ( 3 x ) 2 3
2 4. Simplify the following using the rules of exponents. a ( x y ) ( 2 xy ) b. a b a b c ( ) ( 4 ) 5. It is possible to compose more than two functions. If f( x) = x 1, g( x) = 7 2x and hx = x find 2 ( ) 3, a. f( g( h(1))) b. g( f( h(1))) 6. Using the graph to the right, estimate y a. f(g(5)) b. g(f (3)) f c. f(g(3)) g d. g(f(0)) 7. Given f( x) = 4x 9 and g( x) = 10 3 x, what input to the composition function f ( gx ( )) will result in an output of 7?
3 8. A person who is exercising should not exceed his/her upper heart rate limit. The following table shows the relationship between a person s maximum heart rate and his/her upper heart rate limit. Max. Heart Rate (in beats per min) x Upper Heart rate limit (in beats per min) f (x) a. Which is the input variable? b. Is this a linear function? If yes, how do you know? c. Using function notation, write in symbolic form the relationship between the two variables using x and f (x) as described in your table. d. The maximum heart rate is determined by subtracting the age of the person from 220. Since your maximum heart is given in the table, you will be looking for the age of the person in each case. Using g(a) to represent the maximum heart rate and a to represent the person s age, represent this relationship using function notation in symbolic form.
4 e. Use the relationship you found in part d to determine the age for each category in your table. Age (in years) Max. Heart Rate Upper Heart Rate (beats per min.) Limit (beats per min.) f. Using the composition of the two functions, find one function f (g(a)) that will enable you to use age to determine upper heart rate limit. g. Find f (g(80)). What does the input represent? What does the output represent? h. What is a realistic domain for the composition of the functions? 9. The total cost of a lunch including tip and sales tax is a function of the price of the lunch. The tip and sales tax in the amount of 23% of the cost of the lunch must be added to the cost of the lunch to get that total cost. a. Using function notation g (x) to represent the cost of the lunch, including tip and tax, and x to represent the cost of the lunch, write the function that represents the total cost of the lunch in symbolic form. b. Create a table of values using the inputs in the table. Cost of lunch in dollars x Total cost in dollars including tip and tax g(x)
5 c. Graph the function on the grid below. Make sure your scales for the x-axis and y-axis are the same. Label each axis. d. If we interchange the input and output, then the cost of the lunch will be a function of the cost of the lunch including the tax and the tip. Create a table of values showing this interchange. Input, x Output, h (x) e. Write the function h(x) in symbolic form. f. If you have $16 in your pocket, how much can you afford for your lunch? (Remember you will tip and tax a total of 23%) g. Determine the composition of the two functions, g(h(x)). Notice from the composition of the functions that the two functions are inverses because one function "undoes" the other to give the same value for the output that you first put into the composition. h. Graph this function h(x) on the same grid as you used to graph g(x) on the preceding page. You can show that two functions are inverses if their graphs are reflected about the line y=x. Is this true for these functions? i. Determine g(h(12)). Explain how you arrive at your answer. j. What is another symbol that could be used to indicate the inverse of the function g(x) other than naming it h(x).
6 Answers to CE 1010 Compositions of Functions HW 1. a. 3 b. 3 c a. 5 b. 34 c. 3x + 14 d. 3x a. y = x 20 b. y = 4x 6 c. y = -27x 6 4. a. -8x 9 y 14 b. 27 c. a 17 8 b 5. a a. 0 b. 3 b. 2 c. 2 d x = 2 8. a. Maximum Heart Rate b. Yes, the rate of change is constant. c. f (x) = 0.85x d. g(a) = 220 -a e. Age (in years) in table: 20, 30, 40, 55, 70 f. f (g(a)) = a g. f (g(80)) = 119 Input is age. Output is upper heart rate limit. h. 0 < a 90 Answers will vary. 9. a. g(x) = 1.23x b. Cost of lunch in dollars x Total cost in dollars including tip and tax g(x) c. Includes graph from part h. d. Input, x Output, h(x)
7 e. h(x) = ' (.*+ f. $13 for lunch g. g(h(x)) = x and h(g(x))=x h. The two functions are symmetrical about the line y = x. See graph in part C i. g(h(12)) = 12 If the two functions are inverses, then the input 12 will produce an output of 12. j. g -1 (x) Copyright 2016 Pearson Education, Inc.
Section 4.1 Polynomial Functions and Models. Copyright 2013 Pearson Education, Inc. All rights reserved
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Name Date Cumulative Test 1 Evaluate the expression. 1. 7 + 6 3. 4 5 18 3. 4[3 (17 1) ] 4. 3 [(5 + 3) 31] 5. 3(5m 4) when m = 6. 9x 4 when x = 3 Write an algebraic expression, an equation, or an inequality.
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