Functions 3: Compositions of Functions

Transcription

1 Functions : Compositions of Functions 7 Functions : Compositions of Functions Model : Word Machines SIGN SIGNS ONK KNO COW COWS RT TR HI HIS KYK KYK Construct Your Understanding Questions (to do in class). Describe the effect of a) Machine on a word b) Machine on a word.. Consider a combination of Machines and such that the output of is used as the input for. See Eample below, left. This is called a composition of these machines. a. Perform the same series of operations using NIP as the input (below, right). Eample POT POTS STOP NIP b. convention, the composition contained within the dotted bo above is called (which is read composed with ). Label both dotted boes above with the caption Machine. c. For the composition in part a., which machine appears in the name first or [circle one]? d. Which machine is evaluated first, or [circle one]?. Using the naming convention from above, construct a name for this composition (shown two times below) and label each dotted bo below with this name. POT NIP a. Fill in the missing outputs of this machine. b. Does Machine (in this question) produce a different output from Machine (found in the previous question)? In other words, is the order of operation important?

2 8 Functions : Compositions of Functions Model : Composition of Functions It is possible to compose functions as we composed machines on the previous page. The result, a composite function, can be written as shown below, or using the smbol. g f g( ) f( g( )) Composition of Two Functions Composite Function Construct Your Understanding Questions (to do in class). ased on the naming conventions from the previous page, which is the correct name of the composite function in Model? f g or g f [circle one and eplain our reasoning] 5. ssume f ( ) and g( ) a. What is the output of g f with an input of? This can be written ( g f)() b. What is the output of f g with an input of? This can be written ( f g)() c. You can arrive at an algebraic epression for a composition b using as the input. Fill in each of the following boes to determine ( g f)( ), and then ( f g)( ). The first bo is filled in for ou. f g ( g f)( ) g f ( f g)( ) d. For f and g as defined in this question, does f g g f? Eplain.

3 Functions : Compositions of Functions 9 6. (Check our work) Substitute into the equation ou derived a. for ( g f)( ) and check that our answer is 0. b. for ( f g)( ) and check that our answer is 6. Use this to also check our answers to parts a) and b) of Question 5. If an of our group s answers do not fit, check our methods against those of another group. 7. The function f( ) can be epressed as a composition of two simpler functions, h and g where h ( ) and g( ). For these functions f, g, and h, does f ( ) ( g h)( ) or f ( ) ( h g)( ) [circle one]? 8. Consider the function ( g h)( ) f( ) ( ). f() ( ). Propose two functions g and h such that 9. Later in this course we will discover that it is ver useful to decompose a function into simpler functions (as ou were asked to do in the previous two questions). For the following list of functions, propose two functions g( ) and h ( ) such that f ( ) ( g h)( ). You ma not use the functions f ( ) or g ( ). (Construct an eplanation for wh using the function would make this question too eas.) a. f() ( ) d. f( ) b. f ( ) 7 e. f ( ) c. f ( ) f. f( )

4 0 Functions : Compositions of Functions Model : Shifting t right is the parabola h ( ), along with four related parabolas, each one identical to h ( ), but shifted b a distance of units up, down, right, or left, respectivel Construct Your Understanding Questions (to do in class) 0. In Model, label the graph of h ( ) and the graphs of the four other functions listed in the table below. Check our work b finding the verte of each parabola and confirming that this point is satisfied b the equation ou chose as a match. a. Complete Column b filling in the word LEFT, RIGHT, UP, or DOWN. b. Complete Column : Each shift of h ( ) can be represented as a composition of the original function h ( ) and another function g(). For each row, propose a function g() that accomplishes this shift. [The first one is done for ou.] c. In Column indicate if h g or g h give the function in Column. Function Direction of Shift from h ( ) g() that can accomplish this shift h g or g h [indicate which] gives f() shown in Column f() = + g( ) f() = f() = ( + ) f() = ( ). (Check our work) f ( ) shown at right is the graph of a sine wave shifted up b units. ssume h ( ) sin. Which function g() from the previous question and which composition h g or g h or both [circle all that appl] will give the function f ( ) shown at right? = f ()

5 Functions : Compositions of Functions. (Check our work) In general, if ou want to shift a function h ( ) a vertical distance c (down or up) to produce a new shifted function f ( ), ou can compose h ( ) with another function g( ) c to give f ( ). a. (Check our work) Check that this information is consistent with our answer to the previous question. b. c in the function g( ) c can be positive or negative. Describe the result of the composition when c is positive versus negative. c. Which composition h g or g h or both [circle all that appl] will give the function f ( ), as described above?. Suppose ou want to shift a function h ( ) a horizontal distance c (left or right) to produce a new shifted function f ( ). Write a function g( ) and an epression stating how h ( ) and g( ) can be composed to give f ( ). [Hint: look back at the horizontal shifts in Question 0.]. ssume: h ( ) gv ( ) and g ( ) a. Describe the effect of the composition gv h gh in terms of a) horizontal and b) vertical shifting of the graph of the parabola in comparison to the original function h. h b. (Check our work) Is our answer to part a) consistent with the fact that this shifted parabola will have its verte at the point (,)? c. Evaluate the composition ( gv h gh)( ) so as to generate the equation in terms of for this shifted parabola. Remember to evaluate the functions in the name from right to left (i.e., evaluate g h first).

6 Functions : Compositions of Functions 5. (Check our work) Is our answer to the previous question consistent with Summar o F.? Summar o F.: Shifting as Described b Compositions of Functions When gh( ) ch and gv( ) cv the composition gv h gh shifts the function h ( ) in the horizontal direction b an amount c h in the vertical direction b an amount c v 6. In Summar o F. a. when c h is positive, the horizontal shift is right or left [circle one]. b. when c v is positive, the vertical shift is up or down [circle one]. Model : Reflecting bout the and es Etend Your Understanding Questions (to do in or out of class) 7. Identif the graph of the function f( ) in Model. a. Which line is a reflection of f ( ) about the ais: The light solid line or the dashed line [circle one] in Model? b. Which line is a reflection of f ( ) about the ais: The light solid line or the dashed line [circle one] in Model? c. Write an equation for each of these lines.

7 Functions : Compositions of Functions 8. (Check our work) re our answers on the previous page consistent with the following? The functions f( ) and f( ) are reflections of each other about the ais. The functions f( ) and f( ) are reflections of each other about the ais. 9. The functions f () and f ( ) are shown in bold on the right of Model. a. Identif the equations of the light solid curve, and the dotted curve on this graph. b. The reflection of f ( ) about the ais is not shown in Model. dd it and label it with its equation. 0. function f ( ) can be composed with the function g( ) to give a reflection of f ( ). a. Which composition ( f g or g f ) gives the reflection of f ( ) about the ais? [circle one] b. Which composition ( f g or g f ) gives the reflection of f ( ) about the ais? [circle one]. (Check our work) Check our conclusions above with at least one other group, then complete Summar o F. b filling in each blank with g f or f g, as appropriate. Summar o F.: Reflecting of Functions For a function f ( ), a composition with the function g( ) results in a reflection about the ais when the composition is ais when the composition is

8 Functions : Compositions of Functions Model 5: Stretching and Compressing of Functions Etend Your Understanding Questions (to do in or out of class). Each graph in Model 5 shows the function h ( ) sinin bold. Determine which graph in Model 5 shows h ( ) along with versions of h ( ) that are stretched and compressed in. a. the vertical direction. (label this graph vertical stretch/compress ) b. the horizontal direction. (label this graph horizontal stretch/compress ). Complete the table in Summar o F. b writing the correct pair of terms in each empt bo. Choose from: compressed verticall, compressed horizontall, stretched verticall, stretched horizontall. Summar o F.: Stretching and Compressing of Functions For a function h, ( ) a composition with the function g( ) c(where c is positive) results in stretching or compressing of the original function to give a new function f ( ). 0 c (e.g. ½) c > (e.g. ) f g h Compared to h, ( ) f ( ) is Compared to h, ( ) f ( ) is f h g Compared to h, ( ) f ( ) is Compared to h, ( ) f ( ) is

9 Functions : Compositions of Functions 5 Confirm Your Understanding Questions (to do at home). Use a graphing program to plot the graph of the function f ( ) a where a. Then also plot the graph of the function f ( ) a where the value of a is given below. You will have graphs when ou are finished. For each value of a, describe in words the effect of changing a from a to a. a b. a c. a 0 5. Use a graphing program to plot the function f () ( c) where c 0 and the changes noted below. For each change, describe in words the effect of changing c from c 0 to a. c b. c c. c 0 6. Use a graphing program to plot the function f () d where d 0 and the changes noted below. For each change, describe in words the effect of changing d from d 0 to a. d b. d c. d 0 7. Write the equation the parabola indicated b each arrow. f () (shown below) shifted a distance c in the direction

10 6 Functions : Compositions of Functions Notes