- Strisuksa School - Roi-et. Math : Functions

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1 8 EP-Program - Strisuksa School - Roi-et Math : Functions Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 00 :Wattana Toutip wattou@kku.ac.th 8 Functions 8. Composition and inverses of functions A function y f ( ) can be written in the form f : y. The set of possible values is called the domain, and the set of possible y values id the range or image or co-domain. The composition f g or fg of two functions f and g is defined by : f g( ) f { g( )} The identity function iis ( ) such that i( ) for all. The inverse of a function f is the function f { f ( )} f { f ( )} Equivalently, 8.. Eamples f f f f i.. Let f and g be defined by f such that : f :, g : 5. Find (i) f g() (ii) g f() (iii) f g (iv) Solution (i) Apply g first and then f. g() 8, f ( 8) 5. f g() 5 (ii) Apply f first and then g. f () 5, g(5). g f() (iii) g( ) 5.Apply f ; f ( 5 ) ( 5 ) (iv) f g : 5 5 f. f is obtained by multiplying by and subtraction.the inverse is obtained by adding and dividing by. f g : ( ). The function h is defined by h : 4.Its domain is the set of all real numbers. Find the range of h, and eplain why it has no inverse. Restrict the domain of h so that it does have an inverse. Solution

2 is always at least 0. Hence The range of h is { y: y 4}. h() h( ).Hence h () does not eist. h 4 is always at most 4. cannot be defined on, for eample. If the domain of h is restricted to positive values of, then the inverse is h :. Restrict h to the domain { : 0} 8.. Eercises. With f and g as in Eample above, find : a) g f( ) b) f g( ) c) g f d) g. Let j : and k : 4.Find: a) jk ( ) b) k j( ) c) jk d) k j e) f) g) j k ( jk ) h) k j i) j) j k k j. For the functions f and g of Eample above show that: f g g f g f f g and 4. The function m is defined by m : 5,for.Find the range of m. 5. Find the inverse of m of Question 4, and find its domain and range. 6. Find the ranges of the following functions: a) sin b) c) cos d) 0 e) sin f) g) h) ( ) 5 4 ( )

3 7. For the following functions, state whether or not they have an inverse.find the inverse if does eist. and for those functions without an inverse restrict the domain so that an inverse eists. a) b) 4 c) d) sin e) f) ( ) 8. Graphs of functions Standard graphs are shown in Fig 8. to8.7. Fig 8. sin Fig 8. cos

4 Fig 8. tan Fig 8.4 Fig 8.5 Fig 8.6 e Fig 8.7 ln Transformation of graphs We can convert these standard graphs to more complicated graphs, such as sin, or cos, or. Suppose we have the graph of y f ( ).The effect of simple transformations on graphs is as follows.in the diagrams the dotted graph is of y f ( ).the filled-n line is of the transformed graph. The graph of y f ( ) a is obtained by shifting up the graph of y f ( ) by a units. (Fig 8.8)

5 Fig 8.8 Fig 8.9 The graph of y f ( a) is obtained by shifting the graph of y f ( ) a units to the left. (Fig8.9) The graph of y af ( ) is obtained by stretching the graph of y f ( ), along the y -ais by a factor of a. (Fig8.) The graph of y f ( a) is obtained by contacting the graph of y f ( ), in the line y. (Fig 8.) Fig 8.0 Fig8.

6 Fig 8. Fig 8. Asymptotes If a graph approaches a straight line, as y or tends to infinity, then the straight line is an asymptote of the graph.(fig 8.) Odd, even, Periodic functions An even function is one for which f ( ) f ( ).The graph of an even function is symmetric about the origin. (Fig 8.5) An function is periodic with period a if f ( a) f ( ) for all. The graph repeats itself after a units along the - ais. (Fig 8.6) Fig Eamples. Using the graph of y sin sketch the graph of y sin, for in the range Solution The graph of sin is the dotted line of Fig 8.7. Contract it by in the direction and shift it down by.the result is the filled in line of Fig 8.7.

7 Fig 8.7 Fig 8.8.Let a function be defined by y Find the equations of its asymptotes. Draw the graph of the function. Solution Use long division, as in Chapter, to rewrite 5 y As tends to infinity, then y tends to. The asymptotes are y and A graph of the curve is shown in Fig Eercise. sketch the graphs of the following functions, for in the range 0 60 a) cos b) sin c) cos d) sin e) sin( 80 ) f) cos( 45 ) g) tan( 90 ) h) sin( ) i) ln( ) j) e. sketch the graphs of the following :

8 a) b) c) d) e) f). Find, where relevant, the equations of the asymptotes of the graphs in Question. 4. Sketch, on the same paper, the graphs of between the two graphs? 5. Sketch the following graphs: a) b) c) d) e) sin f) g) h) e and ln. What is the relationship 6. Sketch the following graphs, and in each case write down the equation of the line of symmetry: a) b) 4 c) 5 d) e) f) Each of the following graphs is of he form y b c. In each case find the values of b and c.

9 (a) (b) Fig 8.9 (c) 8. Each if the following graphs is of the form of b and c (a) y b c.in each case find the values (b)

10 Fig8.0 (c) 9. Each if the following graphs is of the form of a and b. y a b.in each case find the values (a) (b) Fig 8. (c)

11 0. Sketch the graphs of the following functions: a) b) c) d) e) f) g) h). Find the equations of the asymptotes of the graph in Question 9.. Draw the graph of 0.. Draw the graph of y.by drawing a suitable line solve the equation y. By drawing a suitable line solve the equation Draw on the same paper the graphs of y sin cos, for Draw on the same paper the graphs of y tan and y sin, for 0 60.Hence solve the equation tan sin.

12 6. By drawing suitable graphs solve the following equations. The trigonometric equations are to be solved in the range a) cos sin b) e c) sin cos d) tan sin 7. For each of the following functions, state whether they are odd, even or periodic. If they are periodic give the period. a) b) c) sin d) cos e) sin f) cos g) h) tan i) e j) tan k) sin l) cos( 45 ) 8. Eamination Questions. The functions f and g are defined as follows :. f : and g : (i) Show that gf (5) 8, and write down epression for the following: a) gf ( ) b) fg( ) (ii) If the domain is the set of real number, state the range of gf. (iii) Say whether the inverse of gf is a function, and justify your answer. a) A function is defined by (i) Epress f f : for all real values of ecept. and if in similar form and state the values of for which these functions are not defined. (ii) Epress the equation f ( ) in the form a b c 0 and hence show that there are no values of which map on to themselves under the function.

13 a 4 a) The function g is defined by g: for all real values of ecept. Find the positive value of a for which there is only one value of satisfying g( ).. Using the same scales and aes draw the graphs of y sin and y cos( 60 ) between the values 0 and 60.Take a scale of cm for 0 on the -ais and 4 cm for unit on the y -ais and plot points corresponding to 0 intervals on the - ais. Use your graph to : a) Solve the equation sin cos( 60 ), for 0 60, b) Estimate the range of values of, between 0 and 60, foe which sin cos( 60 ). 4. Draw separate sketches of the curves a) y b) y c) y 5. Epress 6. in the form 9 The graph of y a r s, where arand, s are constants. 9 may be obtained from the graph of y by means of appropriate translations and scalings (stretches). Describe suitable transformations in detail and the order in which they are to be used. The function is defined for the domain by Sketch the graph of y f ( ). State the domain and range of f and determine f ( ) 9. f. Fig 8. the diagram shows the graph of a function f ( ) over the interval 0. You are given that f is an odd, periodic function with period 6.Draw the graph of f over the interval 6 6.

14 What is the smallest positive value of a for which the function defined by f ( a) is an even function? 7. The figure shows a sketch of the part of graph of y f ( ) for 0 a. The line a is a line of symmetry of the graph. Sketch on separate aes the graphs of : a) y f ( ) for 0 4a b) y f ( ) for 0 a c) y f ( ) for 0 a d) y f ( a) for a 4a Fig 8. Common errors. Composition and inverses a) Do not composition with multiplication, or inverse with dividing. f g( ) f ( ) g( ), and b) When finding f g( ) f ( )., f must be applied to the whole of g.if ( ) for eample f : and g :, then : f g : ( ) 6 8. c) There is nothing special about the letter.any letter will do to defined a function. The function f above could be defined as: f : y y, or f : Z Z d) f g means :do g first and then f. Do not get this the wrong way round.. Graphs a) When plotting a curve do not join up the points with straight line segments. Make as smooth a curve as you can between them. b) Changing y f ( ) to y f ( a) moves the graph a units to the left, not to the right. c) Changing y f ( ) to y f ( a) shrinks the graph by a factor of a.it does not epand it. d) An asymptote does the graph into two separate pieces.do not try to make the graph cross the asymptote.

15 Solution (to eercise) 8... (a) 5 (b) 50 (c) 7 5 (d) 5. (a) 7 (b) 7 (c) 6 5 (d) 6 5 (e) 4 (f) 5 (g) 6 5 (h) 6 5 (i) 6 5 (j) 6 4. y: y Dom. : 7, Range y: y 5 (a) (b) : 0 (c) 4 (d) 0 (e) (f) 5 (g) (h) all. (a) No. restrict to 0

16 (b) Yes. 4 (c) Yes. (d) No (e) No. (f) Yes (a) 0, y 0 (b), y 0 (c), y 0 (d) 0, y (e), y (f), y 4.Inverse of each other. 6. (a) 0 (b) 0 (c) (d) (e) (f) 6 (a) b 0, c (b) b, c 0 (c) b, c (a) b, c (b) b 0, c (c) b, c 0 (a) b 0, a (b) a 0, b (c) a, b. (a) 0, y (b) y 0,

17 (c), y ,70,0, (a) 7,07 7. (b) 0.44 (c) 66,04 (d) 0,60,80,00 (a) Even (b) Odd (c) Odd, periodic, period 60 (d) Even, period, 60 (e) Odd (f) Even, periodic,80 (h) Odd, periodic, 80 (j) Odd, periodic, 60 (k) Odd, periodic, 70 (l) Periodic,80 =========================================================== References: Solomon, R.C. (997), A Level: Mathematics (4 th Edition), Great Britain, Hillman Printers(Frome) Ltd. More: (in Thai)