Chapter 2 Functions and Their Graphs

Transcription

1 Chapter Functions and Their Graphs Section (, ) + ( ) ( ) + or or We must not allow the denominator to be + ; Domain: { } > > < Solution set: { < } or (, ) independent; dependent 6 range 7 [, ] We need the intersection of the intervals [,7 ] and [,] That is, domain of f domain of 8 ; f; g 9 ( g f ) or g f g f g f + g False; every function is a relation, but not every relation is a function For eample, the relation + y is not a function True True False; if the domain is not specified, we assume it is the largest set of real numbers for which the value of f is a real number False; the domain of f is { } Function Domain: {Elvis, Colleen, Kaleigh, Marissa} Range: {Jan 8, Mar, Sept 7} 6 Not a function 7 Not a function 8 Function Domain: {Less than 9 th grade, 9 th - th grade, High School Graduate, Some College, College Graduate} Range: {$8,, $,, $6,, $,8, $67,6} 9 Not a function Function Domain: {,,, } Range: {,, 7, } Function Domain: {,,, } Range: {} Function Domain: {,,, } Range: {,, 7} Not a function Not a function Function Domain: {,,, } Range: {,, } 6 Function Domain: {,,, } Range: {,, 6} 7 Copyright 6 Pearson Education, Inc

2 Chapter : Functions and Their Graphs 7 Graph y The graph passes the vertical line test Thus, the equation represents a function + y Solve for y: y ± For, y ± Thus, (, ) and (, ) are on the graph This is not a function, since a distinct - value corresponds to two different y-values 8 Graph y The graph passes the vertical line test Thus, the equation represents a function Graph y + The graph passes the vertical line test Thus, the equation represents a function 9 Graph y The graph passes the vertical line test Thus, the equation represents a function 6 Graph y The graph passes the vertical + line test Thus, the equation represents a function Graph y The graph passes the vertical line test Thus, the equation represents a function y Solve for y: y ± For, y ± Thus, (, ) and (, ) are on the graph This is not a function, since a distinct - value corresponds to two different y-values y ± For, y ± Thus, (, ) and (, ) are on the graph This is not a function, since a distinct - value corresponds to two different y-values 7 + y Solve for y: + y y y y ± For, y ± Thus,, and, are on the graph This is not a function, since a distinct -value corresponds to two different y-values y Solve for y: y ± For, y ± Thus, (, ) and (, ) are on the graph This is not a function, since a distinct -value corresponds to two different y-values 76 Copyright 6 Pearson Education, Inc

3 Section : Functions 8 y Solve for y: y y y ± y For, y ± Thus,, and, are on the graph This is not a function, since a distinct -value corresponds to two different y-values f + 9 a f + b () () () f + + c f + d f + f + + e f f ( ) ( ) ( ) g f + + h f ( h) ( h) ( h) h+ h + + h + 6h+ h + + h f + a f ( ) ( ) + b f () () + c f + d f + f + + e f f ( ) ( ) ( ) g f h f ( h) h ( h) f + ( + ) a + h+ h + + h h h + + h f + b f () + c f d f ( ) + + ( ) + + f + + e f f ( ) g f + h f ( h) ( ) h + h ( + h) + h h 77 Copyright 6 Pearson Education, Inc

4 Chapter : Functions and Their Graphs f + a f ( ) + b f () + c f d f e f ( ) f f ( ) g f + h f ( h) f + ( ) ( ) h + h+ h + + h + + h+ a f ( ) + + b f () + + c f ( ) + + d f ( ) + + f + e f f f + + g h f ( + h) + h + f + a f + b f () + c f + f + d e f + + f f ( + ) ( + ) + ( + ) f + + g h f ( + h) ( + h) + ( + h) f + a f ( ) + h+ h + + h + + () () + + b f () c f d f e f f f ( ) g f h f ( h) h + + h+ + + h + h 78 Copyright 6 Pearson Education, Inc

5 Section : Functions 6 f a ( + ) f ( + ) 8 b f () c f d f e f ( + ) ( + ) ( ) ( ) + ( + ) f f ( ) g f h f ( h) + ( + + ) ( + ) + 7 f ( + ) ( + ) ( + h+ ) Domain: { is any real number} f + Domain: { is any real number} f + Domain: { is any real number} h ±, Domain: { } F + + ( + ), Domain: { } + G ( ),, ± Domain: {,, } h Domain: { } 6 G Domain: { } f + Domain: { is any real number} g f 9 9> > 9 Domain: { > 9} 6 ±, Domain: { } 79 Copyright 6 Pearson Education, Inc

6 Chapter : Functions and Their Graphs 8 f > 9 > Domain: { > } p > > Domain: { > } 6 q 6 6 Domain: { } t Pt () t t t Also t t t t 7 Domain: { tt, t 7} hz z + z z + z Also z z Domain: { zz, z } 6 f + g a ( f + g) Domain: { is any real number} b ( f g) (+ ) ( ) c d Domain: { is any real number} ( f g) (+ )( ) Domain: { is any real number} f + g Domain: e ( f + g)() () f ( f g)() + 7 g h ( f g)() 6() f () () 7 g () 6 f + g a ( f + g) + + Domain: { is any real number} b ( f g) (+ ) ( ) c d Domain: { is any real number} ( f g) (+ )( ) Domain: { is any real number} f + g Domain: 8 Copyright 6 Pearson Education, Inc

7 Section : Functions 6 e ( f + g)() () f ( f g)() + g h ( f g)() 6() 6() f () + + () g () f g a b c d e f g h ( f + g) + + Domain: { is any real number} ( f g) ( ) ( ) + Domain: { is any real number} ( f g) ( )( ) Domain: { is any real number} f g Domain: { } ( f + g)() () + (9) ( f g)() () + (6) ( f g)() () () (8) () f () g () () 66 f + g + a ( f + g) Domain: { is any real number} b ( f g) ( + ) ( + ) Domain: { is any real number} c ( f g) ( + )( + ) d e f g h Domain: { is any real number} f + g + + Domain: ( f + g)() () + () + (7) + (9) ( f g)() () + () + (6) + (6) ( f g)() 8() + () + () + 8() + (8) + () f () + () + + () g () + () f g a ( f + g) + Domain: { } 8 Copyright 6 Pearson Education, Inc

8 Chapter : Functions and Their Graphs b ( f g) ( ) + Domain: { } h f () g c ( f g) ( ) Domain: { } f d g and Domain: and e ( f + g)() + () f ( f g)() () + + g ( f g)() () h 6 f () g () 68 f g a ( f + g) + Domain: { is any real number} b ( f g) Domain: { is any real number} c ( f g) Domain: { is any real number} f d g Domain: { } e ( f + g)() f ( f g)() g ( f g)() 69 f + g a b c d e ( f + g) Domain: { } ( f g) + Domain: { } ( f g) + + Domain: { } + + f + + g Domain: { } ( f + g)() + f ( f g)() g ( f g)() + + () f h () + g 7 f g a ( f + g) + and and Domain: { } b ( f g) and and Domain: { } 8 Copyright 6 Pearson Education, Inc