Unit 4 Day 8 Symmetry & Compositions

Transcription

1 Unit 4 Day 8 Symmetry & Compositions

2 Warm Up Day 8 1. f ( ) 4 3. g( ) 4 a. f(-1)= a. -g()= b. f(3)= b. g(+y)= c. f(-y)= c. g(-)= 3. Write and graph an equation that has the following: -Nonremovable discontinuity at 5 -Removable discontinuity at -3 -Horizontal asymptote of y=5/

3 Warm Up Day 8 ANSWERS 1. f ( ) 4 a. f(-1)= 3-4. g( ) 4 a. -g()= b. f(3)= c. f(-y)= y 3 b. g(+y)= + y + 4y 4 4y c. g(-)= Write and graph an equation that has the following: -Nonremovable discontinuity at 5 -Removable discontinuity at -3 5( 3)( f ( ) -Horizontal asymptote of y=5/ ( 3)( 5 1) )

4 Homework Questions? Packet p. 6-7

5 Announcements Quiz Unit 4 #1 Corrections due Thurs 3/ If you complete those corrections & do better on Unit 4 Test, the Unit 4 Test grade can replace that quiz Grade Tutorials for credit due Friday 3/3! Tutorials are Most mornings at 7 AM Monday & Wednesday 1 st half lunch Make-Up work due Monday 3/6! Reminder: Midterm is Tuesday 3/7 Midterm Packet due!

6 Notes: Symmetry, Even & Odd Functions, Function Composition

7 Symmetry: About the y-ais Eample: f ( ) Graph: A function is EVEN if it is symmetrical about the y-ais. NON-Eample: f ( ) ( 5) Graph:

8 Symmetry with respect to the y-ais A.k.a. Reflection about y-ais To prove a function is even algebraically, show that E: where n is even

9 Determining Algebraically if a function is Even, Odd, or Neither Prove whether f() = 4 or neither. is even, odd, We ll check if

10 Determining Algebraically if a function is Even, Odd, or Neither Prove whether f() = 4 or neither. is even, odd, We ll check if 4 4 f ( ) ( ) f ( ) f ( ) f ( ) is even

11 Symmetry Checking in Table Note: You can often double-check if a function is even in the table Eample: f ( ) BUT To prove Even functions you must show Algebraically that

12 Symmetry: -ais A.k.a. Symmetric with respect to the -ais. Graphs with this symmetry are not usually functions. They are mirrors. (,-y) mirrors (,y)

13 Symmetry: About the origin Eample: f ( ) 3 Graph: A function is ODD if it is symmetrical about the origin. NON-Eample: f ( ) 3 1 Graph:

14 Symmetry about the Origin A.k.a. Symmetric with respect to the origin. Can be seen as Rotation about origin 180. To prove odd functions algebraically show that : E: where n is odd.

15 Determining Algebraically if a function is Even, Odd, or Neither Prove whether f() = 5 neither. is even, odd, or 1 st we ll check if nd we ll check if

16 Determining Algebraically if a function is Even, Odd, or Neither Prove whether f() = 5 neither. is even, odd, or 1 st we ll check if f ( ) ( ) 5 5 f ( ) f ( ) f ( ) is NOT even nd we ll check if 5 5 f ( ) ( ) f ( ) f ( ) f ( ) is odd

17 Symmetry Checking in Table Note: You can often double-check if a function is odd in the table Eample: f ( ) 3 BUT To prove Odd functions you must show Algebraically that

18 Is the function symmetric? If so, determine the symmetry. Graph Algebraic

19 Is the function symmetric? ANSWER If so, determine the symmetry. Graph Algebraic f ( ) f ( ) Even Symmetric about the y ais

20 Is the function symmetric? If so, determine the symmetry. g( ) 3 3 Graph Algebraic

21 Is the function symmetric? ANSWER If so, determine the symmetry. g( ) 3 3 Graph Algebraic g( ) g( ) Odd Symmetric about the origin

22 Is the function symmetric? If so, determine the symmetry. h( ) 7 Graph Algebraic

23 Is the function symmetric? ANSWER If so, determine the symmetry. h( ) 7 Graph Algebraic h( ) h( ) h( ) h( ) Neither

24 You Try! Is the function symmetric? If so, determine the symmetry. 3. g ( ) f( ) f( ) 3

25 You Try! ANSWER Is the function symmetric? If so, determine the symmetry. 3. g ( ) 3 9 g( ) g( ) Odd 5. f( ) 4. f( ) f ( ) f ( ) f ( ) f ( ) Neither f ( ) f ( ) Even

26 Now let s combine functions Recall function notation!

27 Let s step it up a notch! If f()= 3, (Simplify when possible) Find: f(4) f() Find: f(+1) Find: f(+h) f()

28 Let s step it up a notch! If f()= 3, (Simplify when possible) Find: f(4) f() Find: f(+1) Find: f(+h) f() = h 3 = 3h

29 Combinations of Functions When you add, subtract, multiply, or divide two functions. Add: Subtract: Multiply: Divide: (f + g) () = f() + g() (f - g) () = f() - g() (f g) () = f() g() (f/g) () = f() / g()

30 Eamples: 3 f( ) g ( ) 3 Add and find the domain: Subtract and find the domain: (f + g) () = f() + g() (f - g) () = f() - g() 3 Multiply and find the domain: (f g) () = f() g() Divide and find the domain: (f/g) () = f()/g()

31 Eamples: Add and find the domain: (f + g) () = f() + g() 3 3 ( f g)( ) 3 Multiply and find the domain: (f g) () = f() g() 3 3 ( f g)( ) 3 3 f( ) g ( ) 3 3 Subtract and find the domain: (f - g) () = f() - g() 3 3 ( f g)( ) 3 D : (,) (, ) D : (,) (, ) Divide and find the domain: (f/g) () = f()/g() ( f / g)( ) D : (,) (, ) D : (, 3) ( 3, ) (, ) 3 3 3

32 Eamples: f ( ) 4 g( ) Add and find the domain: (f + g) () = f() + g() Subtract and find the domain: (f - g) () = f() - g()

33 Eamples: f ( ) 4 g( ) Add and find the domain: (f + g) () = f() + g() ( f g)( ) 4 D :[ 4, ) Subtract and find the domain: (f - g) () = f() - g() ( f g)( ) 4 D :[ 4, )

34 Eamples: f ( ) 4 g( ) Multiply and find the domain: (f g) () = f() g() Divide and find the domain: (f/g) () = f()/g()

35 Eamples: f ( ) 4 g( ) Multiply and find the domain: (f g) () = f() g() ( f g)( ) 4 D :[ 4, ) Divide and find the domain: (f/g) () = f()/g() ( f / g)( ) 4 D :[ 4, ) (, )

36 Compositions Let f & g be two functions such that the domain of f intersects the range of g. (f g)() = f(g()) (g f)() = g(f()) Domain of (f g)() = Intersection of the domain of g() with the domain of f(g()). Domain of (g f)()= Intersection of the domain of f() with the domain of g(f()).

37 Compositions ( ) f g( ) 4 3 (f g)() = f(g()) (g f)() = g(f()) Domain: Domain:

38 Compositions ( ) f g( ) 4 3 (f g)() = f(g()) (g f)() = g(f()) f ( g( )) (4 3 ) g( f ( )) 4 3 Domain: Domain: D : (, ) D :(, )

39 Find the domain of g(f())? f g ( ) 5 ( ) 3 What numbers can t you substitute in to g(f())? Does your domain agree?

40 Find the domain of g(f())? f g ( ) 5 ( ) 3 g f ( ( )) ( 5) 3 D :[ 5, ) Evaluate : g( f ( 9)) What numbers can t you substitute in to g(f())? Does your domain agree?

41 Compositions f ( ) 1 1 g( ) 1 g f g f f g f g Domain: Domain:

42 Compositions f ( ) 1 1 g( ) 1 g f g f f g f g 1 ( f ( g( )) 1 1 ( g( f ( )) 1 ( 1) 1 Domain: D : (,1) (1, ) Domain: D : (, ) (, ) (, )

43 f g ( ) 5 ( ) 3 Evaluate: f ( g() g( 1)) f ( g( h 1))

44 f g ( ) 5 ( ) 3 Evaluate: f ( g() g( 1)) f ( g( h 1)) h h3

45 Decompositions We are now going to undo compositions by breaking a function back into its two pieces so that h()=f(g()). 1. h( ) ( 1) 3( 1) 4 3. h( ) 1

46 Decompositions We are now going to undo compositions by breaking a function back into its two pieces so that h()=f(g()). 1. h( ) ( 1) 3( 1) 4 f ( ) 3 4 g( ) 1 3. h( ) 1 f ( ) 1 g( ) 3

47 Etra Practice up net

48 Warm Up ~ State the following and sketch a graph Domain: Range: & y intercepts: Ma and Min: Increasing: g ( ) 3 Decreasing: Limits at discontinuities: End Behavior using limits:

49 Warm Up ANSWERS ~ State the following and sketch a graph Domain: Range: & y intercepts: NONE Ma and Min: Increasing: Decreasing: (-,0)È(0,1)È(1, ) (-,0)È(0, ) Limits at discontinuities: ma of 3.07 at.4 (0,.4] (,0) [.4,1) (1, ) End Behavior using limits: g ( ) lim f( ) 0 lim f ( ) 1 lim f( ) 0 lim f( ) 0 3 DNE