1.2. Click here for answers. Click here for solutions. A CATALOG OF ESSENTIAL FUNCTIONS. t x x 1. t x 1 sx. 2x 1. x 2. 1 sx. t x x 2 4x.

Transcription

1 SECTION. A CATALOG OF ESSENTIAL FUNCTIONS. A CATALOG OF ESSENTIAL FUNCTIONS A Click here for answers. S Click here for solutions. Match each equation with its graph. Eplain our choices. (Don t use a computer or graphing calculator.). (a) 8 (b) log 8 (c) sin f g h. f,. f,. f s,. f s, 4. f, 5. f s t t t s t s t t 4. (a) 7 (b) (c) (d) 7 s 4 9 Find f t h.. f, t s, h g F G f 7. f, t, h 8. f 4, t 5, h s 9. f s, t, h s Epress the function in the form f t.. F 9 5. u t tan t Copright, Cengage Learning. All rights reserved. 8 Graph each function, not b plotting points, but b starting with the graph of one of the standard functions given in Section., and then appling the appropriate transformations.. 4. cos 5. tan. s 7. cos sin.. sin. 4. s 4 5. s cos 9 5 Find the functions f t, t f, f f, and t t and their domains. 9. f s, t. Suppose we are given the graphs of f and t, as in the figure, and we want to find the point on the graph of h f t that corresponds to a. We start at the point a, and draw a vertical line that intersects the graph of t at the point P. Then we draw a horizontal line from P to the point Q on the line. (a) What are the coordinates of P and of Q? (b) If we now draw a vertical line from Q to the point R on the graph of f, what are the coordinates of R? (c) If we now draw a horizontal line from R to the point S on the line a, show that S lies on the graph of h. (d) B carring out the construction of the path PQRS for several values of a, sketch the graph of h. R Q a a S P f g

2 SECTION. A CATALOG OF ESSENTIAL FUNCTIONS. If f is the function whose graph is shown, use the method of Problem to sketch the graph of f f. Start b using the construction for a,.5,,.5, and. Sketch a rough graph for. Then use the result of Eercise in Section. to complete the graph. Copright, Cengage Learning. All rights reserved.

3 SECTION. A CATALOG OF ESSENTIAL FUNCTIONS. ANSWERS E Click here for eercises. S Click here for solutions.. (a) t (b) h (c) f.. (a) G (b) F (c) t (d) f. π π 7π sin - π π (, ) π Ł(/) (_, ) -œ + _π _π _π π π π f t f s,,, t f,, f f ss,, t t 4,,. f t, t f, f f, t t ,,. Copright, Cengage Learning. All rights reserved.

4 4 SECTION. A CATALOG OF ESSENTIAL FUNCTIONS. f t, t f, f f, t t,. f t s,, t f s s, [ s, ] [, s] f f s, (, s] [s, ) t t s s,,. f t s s,, t f s,, f f s 9,, t t s s,, 4 4. f t, t f 5 4 f f, 4 5 t t,, 4 4,,,,,,, f t s 4,, 4, t f,, 4 s f f 4,, t t 4 8,,. f t h s 7. f t h f t h s s. t 9, f 5. t t t, f t tan t. (a) P a, t a, Q t a, t a (b) (d). f t h (s 5) 4 f Ï S h g R Q P a t a, f t a Copright, Cengage Learning. All rights reserved.

5 SECTION. A CATALOG OF ESSENTIAL FUNCTIONS 5. SOLUTIONS E Click here for eercises.. (a) The graph of 8 mustbethegraphlabelledg, because g is the graph of a power function of even degree, asshowninfigure 7. (b) The graph of log 8 must be the graph labelled h, because h is a graph similar to the graphs of logarithmic functions shown in Figure 4 4. (c) The graph of +sin must be the graph labelled f, because f is the graph of a periodic function.. (a) The graph of 7 must be the graph labelled G, because G passes through the origin. (b) The graph of 7 must be the graph labelled F, because F appears to be an eponential function with -intercept, increasing, and horizontal asmptote. (c) The graph of / must be the graph labelled g, because g has a vertical asmptote at. (d) The graph of 4 must be the graph labelled f, because f has domain [, ). +: Start with the graph of. and shift units to the left. 7. cos(/): Start with the graph of cosand stretch horizontall b a factor of. + + ( + + ) 8. +( +) +: Start with the graph of,shift unit left, and then shift units upward.. /: Start with the graph of / and reflect about the -ais. 4. cos : Start with the graph of cos, reflect aboutthe -ais, and then shift units upward. 9. : Start with the graph of / and shift units to the right. Copright, Cengage Learning. All rights reserved. 5. tan: Start with the graph of tanand compress horizontall b a factor of.

6 SECTION. A CATALOG OF ESSENTIAL FUNCTIONS. sinπ: Start with the graph of sin,compress horizontall b a factor of π, stretch verticall b a factor of,andthenreflectaboutthe-ais : Start with the graph of, shift 4 units to the left and compress verticall b a factor of, andthenshift units downward. sin ( ). π : Start with the graph of sin, shift π units to the right, and then compress verticall b a factor of. 5. +: Start with the graph of, reflect about the -ais, shift unit to the left, and then shift units upward. +. : Start with the graph of /,shift unit + left, and then shift units upward.. ( ) +: Start with the graph of,shift unit to the right, and then shift units upward. Copright, Cengage Learning. All rights reserved ( + ) ++ ( ) +: Start with the graph of, shift unit right, reflect about the -ais, andthenshift units upward.

7 SECTION. A CATALOG OF ESSENTIAL FUNCTIONS : Start with the graph of,shift unit downward, and then reflect the part of the graph from to about the -ais. cos : Start with the graph of cos and reflect the parts of the graph that lie below the -ais about the -ais. ( ) / ( ) (g f)() g / ( ) +, D {, }. ( ) (f f)() f D {, }. ( ) (g g)() g + D {, }. / ( ), ( ) / ( +) ( ) / ( +)+, f ()., D (, ] [, ); g (), D (, ]. (f g)() f (g ()) f ( ) ( ). To find the domain of (f g)(), we must find the values of that are in the domain of g such that g () is in the domain of f. Insmbols,wehave D { (, ] (, ] [, ) }. First, we concentrate on the requirement that (, ] [, ). Because, Copright, Cengage Learning. All rights reserved. f () 9., D [, ); g (), D R. (f g)() f (g ()) f ( ), D { R g () [, )} (, ] [, ). (g f)() g (f ()) g ( ) ( ), D [, ). (f f)() f (f ()) f ( ), D { [, ) } [, ). (g g)() g (g ()) g ( ) ( ) 4, D R.. f () /, D { }; g () +, D R. (f g)() f (g ()) f ( + ) / ( + ), D { + } { }. (g f)() g (f ()) g (/) / +/, D { }. (f f)() f (f ()) f (/) /, D { }. (g g)() g (g ()) g ( + ) f (). D { }. ( ) (f g)() f + ( + D { }. ( + ) + ( + ) , D R., D { }; g () +, ( + ), ) is not in (, ]. If is in [, ), then we must have. Combining the restrictions and (, ],we obtain D (, ]. (g f)() g (f ()) g ( ), D { (, ] [, ) (, ] }. Now. Combining this restriction with (, ] [, ), weobtain D [, ] [, ]. (f f)() f (f ()) f ( ) ( ), D { (, ] [, ) (, ] [, ) }. Now or. Combining this restriction with (, ] [, ), weobtain D (, ] [, ). (g g)() g (g ()) g ( ), D { (, ] (, ] }. Now. Combining this restriction with (, ], we obtain D [, ].

8 8 SECTION. A CATALOG OF ESSENTIAL FUNCTIONS f (), D R; g ()., D [, ). (f g)() f (g ()) f ( ), D [, ). (g f)() g (f ()) g ( ) /, D [, ). (f f)() f (f ()) f ( ) /9, D R. (g g)() g (g ()) g ( ), D { } [, ]. f () + +, D { } 4. ; g (), D { }. (f g)() f (g ()) ( ) / ( ) + f / ( ) + 4, D { },. (g f)() g (f ()) ( ) + ( +)/ ( +) g + ( +)/ ( +), D { },. (f f)() f (f ()) ( ) + ( +)/ ( +)+ f + ( +)/ ( +) , D {, } 5 4. (g g)() g (g ()) ( ) / ( ) g / ( ), D {, 4}. 4 f () / 5., D (, );g () 4, D R. (f g)() f (g ()) f ( 4 ) / 4, D { 4 > } (, ) (4, ). ( ) (g f)() g (f ()) g 4, (f g h)() f (g (h ())) f (g ( 8. )) f ( 5) ( 5) (f g h)() f (g (h ())) f (g ( )) ( ) f. Let g () 9 and f () 5.Then... (f g)() ( 9) 5 F (). Let g (t) πt and f (t) tant. Then (f g)(t) tanπt u (t). (a) P (a, g (a)) and Q (g (a),g(a)) because Q has the same -value as P anditisontheline. (b) The -value of Q is g (a); this is also the-value of R. The -value of R is therefore f (-value),that is, f (g (a)). Hence,R (g (a),f(g (a))). (c) The coordinates of S are (a, f (g (a))) or, equivalentl, (a, h (a)). (d) We need to plot points onl for the first quadrant since we can see that f is an odd function, and we then know that f f is an odd function, and hence, smmetric with respect to the origin..5.5 f ().5.4 f (f ()) D (, ). (f f)() f (f ()) f D (, ). (g g)() g (g ()) g ( 4 ) ( 4 ) 4 ( 4 ) ( ) /4, / , D R. Copright, Cengage Learning. All rights reserved.. (f g h)() f (g (h ())) f (g ( )) f ( ) (f g h)() f (g (h ())) f ( g ( + )) 7. ( ( f + ) ) / ( + )