6-1 Study Guide and Intervention

Transcription

1 Study Guide and Intervention Arithmetic perations perations with Functions Sum (f + g)() = + g() Difference (f - g)() = - g() Product (f g)() = g() Quotient ( f g )() = g(), g() Eample and g() = -. Find (f + g)(), (f - g)(), (f g)(), and ( g) f () for = + - (f + g)() = + g() Addition of functions = ( + - ) + ( - ) = + -, g() = - = Simplify. (f - g)() = - g() Subtraction of functions = ( + - ) - ( - ) = + -, g() = - = - Simplify. (f g)() = g() Multiplication of functions = ( + - )( - ) = + -, g() = - = ( - ) + ( - ) - ( - ) Distributive Property = Distributive Property = Simplify. ( f g )() = g() Division of functions = + -, - = + - and g() = - Eercises Find (f + g)(), (f - g)(), (f g)(), and ( f g ) () for each and g().. = 8 - ; g() = + 5. = + - 6; g() = -. = - + 5; g() = -. = - ; g() = + - Lesson 5. = - ; g() = + Chapter 6 5 Glencoe Algebra

2 Study Guide and Intervention (continued) Composition of Functions Suppose f and g are functions such that the range of g is a subset of the domain of f. Then the composite function f g can be described by the equation [f g]() = f[g()]. Eample For f = {(, ), (, ), (, ), (, )} and g = {(, ), (, ), (, ), (, )}, find f g and g f if they eist. f[ g()] = f() = f[ g()] = f() = f[ g()] = f() = f[ g()] = f() =, So f g = {(, ), (, ), (, ), (, )} g[ f()] = g() = g[ f()] = g() = g[ f()] = g() = g[ f()] = g() =, So g f = {(, ), (, ), (, ), (, )} Eample Find [g h]() and [h g]() for g() = - and h() = -. [g h]() = g[h()] [h g]() = h[ g()] = g( - ) = h( - ) = ( - ) - = ( - ) - = - 7 = = Eercises For each pair of functions, find f g and g f, if they eist.. f = {(-, ), (5, 6), (, 9)},. f = {(5, -), (9, 8), (-, ), (, )}, g = {(6, ), (, -), (9, 5)} g = {(, 7), (-, 6), (, -), (8, )} Find [f g]() and [g f ](), if they eist.. = + 7; g() = = - ; g() = - 5. = + ; g() = = 5 + ; g() = - Chapter 6 6 Glencoe Algebra

3 Skills Practice Find (f + g)(), (f - g)(), (f g)(), and ( f g ) () for each and g().. = + 5. = + g() = - g() = -. =. = g() = - g() = 5 For each pair of functions, find f g and g f if they eist. 5. f = {(, ), (, -)} 6. f = {(, -), (, ), (, )} g = {(, ), (-, ), (5, )} g = {(-, ), (, )} Lesson 7. f = {(-, ), (-, ), (, )} 8. f = {(6, 6), (-, -), (, )} g = {(, -), (, -), (, -)} g = {(-, 6), (, 6), (6, -)} Find [g h]() and [h g]() if they eist. 9. g() =. g() = - h() = + h() = -. g() = - 6. g() = - h() = + 6 h() =. g() = 5. g() = + h() = + - h() = - If =, g() = +, and h() = -, find each value. 5. f[ g()] 6. g[h()] 7. g[f(-)] 8. h[f(5)] 9. g[h(-)]. h[f()]. f[h(8)]. [f (h g)](). [f (g h)](-) Chapter 6 7 Glencoe Algebra

4 Find (f + g)(), (f - g)(), (f g)(), and ( f g ) () for each and g().. = +. = 8. = g() = - Practice g() = g() = - 9 For each pair of functions, find f g and g f, if they eist.. f = {(-9, -), (-, ), (, )} 5. f = {(-, ), (, -), (, -)} g = {(, -9), (-, ), (, -)} g = {(-, ), (, )} 6. f = {(-, -5), (, ), (, 6)} 7. f = {(, -), (, -), (6, 8)} g = {(6, ), (-5, ), (, -)} g = {(8, ), (-, ), (-, )} Find [g h]() and [h g](), if they eist. 8. g() = 9. g() = -8. g() = + 6 h() = - h() = + h() =. g() = +. g() = -. g() = - h() = h() = + + h() = + If =, g() = 5, and h() = +, find each value.. f[ g()] 5. g[h(-)] 6. h[f()] 7. f[h(-9)] 8. h[ g(-)] 9. g[f(8)]. BUSINESS The function = -. models the manufacturing cost per item when items are produced, and g() = 5 -. models the service cost per item. Write a function C() for the total manufacturing and service cost per item.. MEASUREMENT The formula f = n converts inches n to feet f, and m = f 58 converts feet to miles m. Write a composition of functions that converts inches to miles. Chapter 6 8 Glencoe Algebra

5 NAME DATE PERID Word Problem Practice. AREA Bernard wants to know the area of a figure made by joining an equilateral triangle and square along an edge. The function f(s) = s gives the area of an equilateral triangle with side s. The function g(s) = s s gives the area of a square with side s. What function h(s) gives the area of the figure as a function of its side length s?. PRICING A computer company decides to continuously adjust the pricing of and discounts to its products in an effort to remain competitive. The function P(t) gives the sale price of its Super computer as a function of time. The function D(t) gives the value of a special discount it offers to valued customers. How much would valued customers have to pay for one Super computer?. LAVA The temperature of lava has been measured at up to F. A freshly ejected lava rock immediately begins to cool down. The temperature of the lava rock in degrees Fahrenheit as a function of time is given by T(t). Let C(F) be the function that gives degrees Celsius as a function of degrees Fahrenheit. What function gives the temperature of the lava rock in degrees Celsius as a function of time?. ENGINEERING A group of engineers is designing a staple gun. ne team determines that the speed of impact s of the staple (in feet per second) as a function of the handle length l (in inches) is given by s(l) = + l. A second team determines that the number of sheets N that can be stapled as a function of the impact speed is given by N(s) = s -. What function gives N as a function of l? 5. HT AIR BALLNS Hannah and Terry went on a one-hour hot air balloon ride. Let T(A) be the outside air temperature as a function of altitude and let A(t) be the altitude of the balloon as a function of time. Temperature (ºF) Altitude (km) Altitude (km) a. What function describes the air temperature Hannah and Terry felt at different times during their trip? b. Sketch a graph of the function you wrote for part a based on the graphs for T(A) and A(t) that are given. Temperature (ºF) Time (minutes) 5 6 Time (minutes) Lesson Chapter 6 9 Glencoe Algebra

6 Enrichment Relative Maimum Values The graph of f () = shows a relative maimum value somewhere between f (-) and f (-). You can obtain a closer approimation by comparing values such as those shown in the table. To the nearest tenth a relative maimum value for f () is = Using a calculator to find points, graph each function. To the nearest tenth, find a relative maimum value of the function.. f () = ( - ). f () = - -. f () = f () = Chapter 6 Glencoe Algebra

7 Spreadsheet Activity It is possible to perform operations on functions such as addition, subtraction, multiplication and division. You can use a spreadsheet to investigate the relationships among functions. Consider the functions = +, g() = -, and h() = + +. Find the function values of each function for several values of. Does it appear that + g() = h()? Use Column A for the chosen values of. Columns B, C, and E are, g(), and h() respectively. Use Column D for + g(). For every value of, + g() = h(). Functions.ls A.5 B C g().5 8 D + g() Sheet Sheet Sheet E h() Lesson Eercises Study and use the spreadsheet above.. Find k() = ( + ) + ( - ). How does it compare to h()?. Change the functions in the spreadsheet to =, g() = -, and h() = + -. How are these functions related? Is it true that + g() = h()?. Make a conjecture about (f + g)() for any functions and g().. Make a conjecture about (f - g)() for any functions and g(). Use the spreadsheet to test your conjecture. Does it appear to be true? Eplain your answer. Find (f + g)(), (f - g)(), for each and g(). Use the spreadsheet to find function values to verify your solutions. 5. = = + 7. = g() = 9 + g() = - g() = 6 - Chapter 6 Glencoe Algebra