7. V = khr. 5. r. 7. a =

Transcription

1 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions Section.7 Chec Point Exercises. a. L = N b. L = N c. L = (7) = 68 Sain s moustache grew to 68 inches.. a. W = L b. 7 = (6) = 7 6 c. W = L L W = 7 7 L = 6 6 7( 6) d. W = 6 A 6-foot canoe weighs ounds.. P = D = ( 6) = = 6 P = D P = ( ) = 7. The ressure will be 7. ounds er square inch.. d = v = ( 6) = = 6 8 d = v 8 d = ( ) 6 8 About 6 feet are required.. P = s 9. = 78 = 78 P = s 78 P = P = 6 The new ressure is about.6 ounds er square inch. 6. M = P W 6 = = 8 8P M = W 8 ( ) M = = 8 It will tae minutes. 7. V = hr π = ( )( 6) π π = = 6 πhr V = π( )( ) V = = 96π The volume of the cone is 96π cubic feet. Exercise Set.7. g = h. a = b. r = t 7. a = b 9

2 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr 9. r = s v. s = gt. = x 7 = =. = x = 9 = = 7. W = r = = 9. A = B C = 9 = 9 =. a = bc 7 = 8 6 = 7 =. = x = = 7 = 7x = 7 = 8. = x = = = x = = x 7. = z = = = = = x = z = = = = 6 9. = hxz = = =. =. xz = 8. ( )( ) =. a. L = W b. L.W c. L.() =. Your fingernail length will be. inches. 96

3 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions. C = M = = = C = M C = = 6 The cost is $6.. s = M. = (. ) = 7 s = 7M s = 7(.) = The Blacbird s seed is miles er hour. 7. W = h 7 = 7, = 7 7 =, 7 W = h, 7 W = ( 7 ), W 67 Mr. Wadlow weighed aroximatel 67 ounds. 9. t = r. = = t = r t =. 6 It will tae half an hour.. v = = 8 = 6 6 v = 6 = = 6 = 6. The ressure is 6. ounds. w. i = h = 7 7 = = w i = h 9.( 8 ) i = index: about This erson is not in the desirable range.. I = d = = I = d I =. 6 The illumination is about. foot-candles. 7. e = mv 6 = 8 7 = 6. e. mv e 6.() = 7 The inetic energ is 7 ergs. 97

4 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr 9. c = d ( 8)( 88) 8, = ( 8) ( )( )( 97) c = ( ) c 6 About 6 hone calls er da are made.. 7. Answers ma var. 9. = v = ( v) = v = v The destructive ower is four times as much. v 6. h = r v v h = = r r Reduce the resistance b a factor of. Review Exercises. f( x)= ( x ) + (, ) (, ) 6 a-, b 6 a+, b x axis of smmetr: x =. f ( x) = ( x + ) 6 (, ) ( -, ) ( +, ) x (, ) axis of smmetr: x =. f( x)= x + x+ = ( x x+ )+ + f( x)= ( x ) + (, ) (, ) (, ) (, ) x axis of smmetr: x =. f( x)= x x 6 f( x)= ( x x+ ) 6 ( x ) 8 (, ) (, ) x (, 6) (, 8) axis of smmetr: x =. st ()= 6t + 6t + 8 b 6 t = = = a ( 6 ) It reaches its maximum height after seconds. 98

5 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions s( )= 6( ) + 6( )+ 8 = The maximum height is feet. 6. Vertex (,.) In 98, the divorce was at a maximum with. divorces for ever in the oulation. 7. A = x( x) A = x + x x = = ( ) length = () = The maximum field will have sides of ards and a length of ards for an area of, square ards. 8. f( x)= x + x x The grah rises to the left and falls to the right and goes through the origin, so grah (c) is the best match gx ( )= x 6x + 9x The grah rises to the left and rises to the right, so grah (b) is the best match.. hx ( )= x x + x The grah falls to the left and rises to the right and crosses the -axis at zero, so grah (a) is the best match. f x = x +. () f(x) falls to the left and to the right so grah (d) is the best match.. f( x)=. x +. 78x. x+ 8. Because the degree is odd and the leading coefficient is negative, the grah falls to the right. Therefore, the model indicates that the ercentage of families below the overt level will eventuall be negative, which is imossible.. Nt ()= t + t + Since the degree is even and the leading coefficient is negative, the grah falls to the right. Therefore, the model indicates a atient will eventuall have a negative number of viral bodies, which is imossible.. f( x) = ( x )( x+ ) ( x+ ) x =, multilicit, the grah crosses the x- axis x =, multilicit, the grah touches the x-axis x =, multilicit, the grah crosses the x-axis. f( x)= x x x+ = x ( x ) ( x ) = ( x )( x ) = ( x+ )( x ) x =, multilicit, the grah crosses the x-axis x =, multilicit, the grah touches the x- axis 6. f( x)= x x 9x+ 9 a. Since n is odd and a n >, the grah falls to the left and rises to the right. b. f( x) = ( x) ( x) 9( x) + 9 = x x + 9x+ 9 f( x) f( x), f( x) f( x) no smmetr 99

6 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr c. f( x) = ( x )( x+ )( x ) zeros:,, c. f(x) = (x )(x + )(x + ) zeros: x =,, x x 7. f( x)= x x a. Since n is odd and a n <, the grah rises to the left and falls to the right. b. f( x) = x+ x f( x) = f( x) origin smmetr c. f( x) = xx ( ) = xx ( )( x+ ) zeros: x,, x 9. gx ( )= x + x a. The grah falls to the left and to the right. b. f( x) = ( x) + ( x) = x + x = f( x) -axis smmetr c. x + x x ( x )= x ( x ) ( x+ )= zeros: x =,, 8. f( x)= x + x 8x a. Since h is odd and a n >, the grah falls to the left and rises to the right. b. f( x) = x + x + 8x f( x) f( x), f( x) = f( x) no smmetr. f( x)= x + 6x 9x x a. The grah falls to the left and to the right. b. f( x) = ( x) + 6( x) 9( x) = x 6x 9x f( x) f( x) f( x) f( x) no smmetr

7 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions c. = x ( x 6x+ 9)= x ( x ) ( x )= zeros: x,. f( x)= x x x a. The grah rises to the left and to the right. b. f( x) = ( x) ( x) = x + x f( x) f( x), f( x) f( x) no smmetr c x x x ( x )= zeros: x, x x 7x+. x+ ) x x x+ x + x 7x x 7x 7x x + x + Quotient: x 7x+ x + x x+. x ) x 6x + 7x x + 6x x + 7x x + x x x Quotient: x x+ x. x + x x + ) x + 6x + x x + x 6x x + x 6x + x x x. ( x + x x + 7x+ ) ( x+ ) 7 7 Quotient: x x ( x x x) ( x ) 6 6 Quotient: x + 6x + x+ + x 7. f( x)= x 7x + 9x 7 9

8 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr Quotient: f ( ) = f( x)= x + x x f( x) = ( x )( x + x ) = ( x )( x )( x+ ) Zeros: x =,, 9. x 7x+ 7 6 ( x ) ( x + x )= x = ± 6 + = ± = ± The solution set is, ±,. { }. f( x)= x 6x + x x+ : ±, ± q: ± : ±, ± q. f( x)= x x x + x + x 8 : ±, ±, ±, ± 8 q: ±, ± 8 : ±, ±, ±, ± 8, ±, ±, ±, ± q f ( x)= x + x + x + f( x) has no sign variations, so f(x) has no negative solutions.. f (x) = x x x + x f(x) has sign variations, so f(x) has or ositive real roots. f ( x)= x + x x x f( x) has sign variations, so f(x) has or negative solutions.. f( x)= f ( x)= x + 6x + 8 No sign variations exist for either f(x) or f( x), so no real roots exist.. f( x)= x + x a. : ±, ±, ± q: ± : ±, ±, ± q b. sign variation ositive real zero f( x) = x + x sign variations or no negative real zeros c. is a zero. d. ( x )( x + x+ ) ( x )( x+ ) x = or x = The solution set is{, }. 6. f( x)= 6x + x x +. f( x)= x x 8x + f(x) has sign variations, so f(x) has or ositive solutions.

9 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions a. : ± q: ±, ±, ±, ±6 q : ±, ±, ±, ± 6 b. f( x)= 6x + x x + sign variations; or ositive real zeros. f ( x)= 6x + x + x + sign variation; negative real zero. c. 6 is a zero. 6 6 d. 6x + x x+ = ( x+ )( 6x x+ ) ( x+ )( x )( x ) x = or x = or x = The solution set is,,. 7. f( x)= 8x 6x + 6x a. : ±, ±, ±, ± q: ±, ±, ±, ±8 q : ±, ±, ±, ±, ±, ±, ± 8, ±, ±, ±, ±, ±, 8 ±, ±, ±, ± 8 8 b. f( x)= 8x 6x + 6x sign variations; or ositive real solutions. f( x)= 8x 6x 6x sign variations; no negative real solutions. c is a zero. d. 8x 6x + 6x = x ( 8x x+ ) x ( x 6x+ ) x ( x )( x ) = x = or x = or x = The solution set is,,. 8. f( x)= x x 7x + x + 6 a. : ±, ±, ±, ±6 q: ± : ±, ±, ±, ± 6 q b. f( x)= x x 7x + x + 6 sign variations; or zero ositive real solutions. f( x)= x + x 7x x+ 6 sign variations; or zero negative real solutions. c. 7 6 is a zero. 6 6

10 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr d. x x 7x + x+ 6 ( x+ )( x x x+ ) ( x+ )[ x ( x ) ( x )] ( x+ )( x )( x ) ( x+ )( x )( x )( x+ ) x = or x = or x = or x = The solution set is {,,, }. 9. x + 7x a. : ±, ± q: ±, ±, ± q : ±, ±, ±, ± b. sign variation; ositive real root f( x) = x + 7x sign variation; negative real root c. 7 8 ( x ) ( x + x + 8x+ )= is a zero. a. : ±, ±, ± q: ±, ± q =± ± ± ±,,, b. sign variations; or no ositive zeros f( x) = x x 9x + x+ sign variations; or no negative zeros c. 9 is a zero. ( ) d. f( x) = ( x ) x + x + x f( x) = ( x )( x+ )( x + x ) = ( x )( x+ )( x )( x+ ) x =,,, The solution set is,,,. d. 8. x 7x x + 8x 8 ( x )( x+ )( x + 8) ( x )( x+ )( x + ) x = x =± i The solution set is,, i, i.. f( x)= x + x 9x x is a root and a lower bound

11 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions 6 is an uer bound, but not a zero. : ±, ±, ±, ±, +6, ± q: ±, ± q : ±, ±, ±, ±, ± 6, ±, ±, ± Possible roots are: ±, ±,,, ±, ±. x x x + x+ a. : ±, ±, ±, ±, ±6, ± q: ±, ± q : ±, ±, ±, ±, ± 6, ±, ±, ± b. 6 6 is not a root but is an uer bound. c. is not a root but is a lower bound. d. Possible roots are ±, ±, and ±.. f( x)= x x f ()= () () = f ( )= ( ) ( ) = Continue to use the Intermediate Value Theorem: f(.) =.6 f(.6) =. f(.7). f(.6).9 x.6. f( x)= x + x 8x+ 7 f ( )= ( ) + ( ) 8( )+ 7= f ( )= ( ) + ( ) 8( )+ 7= 7 Continue to use the Intermediate Value Theorem: f(.) =.7 f(.) =. f(.) =.6 f(.).9 x.. (x 6 i)(x 6 + i) = x 6x 6x+ 6 i = x x+ 6 x + x x + 6) x 7x + x + 6 x 8x + x x x + 6 x x + 6 x + x = The solution set is,,. 6± i 6 i 6. (x + i)(x i) = x x + 9i = x x + x x + x x + ) x x + 6x x + x x + x x + 6x x x + x x x x + x x + x x + x = ± () ( ) x = ± i = ± i The solution set is { + i, i, + i, i}.

12 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr 7. (x 7i)(x + 7i) = x 8x = x 8x + 6 x x x 8x + 6) x 7x + 7x 7x 6 x 6x + x x + 7x 7x x + 8x 6x x + 8x 6 x + 8x 6 x x = x = ± ( )( ) x = ± x =, The solution set is,, + 7i, 7i. 8. f( x) = an( x )( x + i)( x i) f( x)= an( x ) ( x x+ ) f()= an( ) [ ()+ ] = an an = f( x)= ( x ) ( x x+ ) f( x)= x x + x x + 8x 6 f( x)= x 6x + x 6 9. f( x)= x i) ( x+ i) ( x+ ) f( x)= an( x + ) ( x + 6x+ 9) f( )= an[ ( ) + ] ( ) + 6( )+ 9 6 = 8an an = f( x)= ( x + ) ( x + 6x+ 9) f( x)= x + 6x + 9x + x + 6x+ 9 f( x)= x + x + x + x+ 8 [ ] ( ). f(x) = a n (x + )(x )(x i)(x + i) f( x)= a n x x 6 ( )( x x + ) x x + x x + f( x)= an x x 6x + x 6 f( x)= an( x x + 6x + x 6) f( )= an ( ) ( ) + 6( ) + ( ) 6 = an an = f( x)= x x + 6x + x 6 [ ]. f( x)= x + x + x : ±, ± q: ±, ± q : ±, ±, ± x + x + x ( x+ )( x x + x ) ( x+ )[ x ( x ) + ( x )] ( x+ )( x )( x + ) x =, x = or x = ±i The zeros are,, ±i. ( ) f( x)= ( x i) ( x + i) ( x + ) x 6

13 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions. gx ( )= x 6x + x + x + 6 : ±, ±, ±, ±8, ±6 q: ± : ±, ±, ±, ± 8, ± 6 q x 6x + x + x+ 6 ( x+ )( x 7x + 8x+ 6) x 9 ( x )( x+ ) x = and x = Horizontal asmtote: n < m, so x ( x+ ) ( x 8x+ 6) ( x+ ) ( x ) x = or x = g(x) = (x + ) (x ). real zeros, one with multilicit two. real zeros; nonreal comlex zeros. real zeros, one with multilicit two; nonreal comlex zeros 6. real zero; nonreal comlex zeros 7. f( x)= x x 9 Smmetr: f ( x) = origin smmetr x-intercet: x = x 9 x x -intercet: ( ) 9 Vertical asmtote: x x 9 = f (x) 7 8. gx ( )= x x + Smmetr: g( x) = x x + g( x) g(x), g( x) g(x) No smmetr x-intercet: x x = -intercet: ( ) ( )+ = Vertical asmtote: x + x = Horizontal asmtote: n = m, so = = x 9. hx ( )= x x x x 6 Smmetr: h( x) = x + x x + x 6

14 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr h( x) h(x), h( x) h(x) No smmetr x-intercets: x x (x )(x + ) x = x = -intercet: ( ) 6 = Vertical asmtotes: x x 6 (x )(x + ) x =, Horizontal asmtote: n = m, so = = 6. rx ( )= x + x + x + ( ) x Smmetr: r( x) = x x + ( x + ) r( x) r(x), r( x) r(x) No smmetr x-intercets: x + x + (x + )(x + ) x =, -intercet: + ( )+ + Vertical asmtote: x + x = Horizontal asmtote: ( ) = n = m, so = = 6. = x x + x Smmetr: f ( x) = x x + f( x) f(x), f( x) f(x) No smmetr x-intercet: x x -intercet: = + Vertical asmtote: x + x = n > m, no horizontal asmtote. Slant asmtote: = x + x + = x x 6. = x + x x Smmetr: f ( x) = x x x f( x) f(x), f( x) f(x) No smmetr 8

15 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions x-intercets: x + x (x + )(x ) x =, -intercet: + ( ) = = Vertical asmtote: x x = Horizontal asmtote: n > m, so no horizontal asmtote. slant asmtote: = x+ + x = x+ 6. f( x)= x x + Smmetr: f ( x) = Origin smmetr x-intercet: x x x x + = f (x) -intercet: = ( ) + = Vertical asmtote: x + x = No vertical asmtote. Horizontal asmtote: n > m, so no horizontal asmtote. Slant asmtote: f (x) = x + x x + = x x 6. gx ( )= x 6x + 6 x Smmetr: g( x) = x + 6x + 6 x g( x) g(x), g( x) g(x) No smmetr x-intercet: x 6x + 6 ( x ) x = -intercet: = ( ) 6( )+ 6 = ( 6 ) Vertical asmtote: x x = Horizontal asmtote: n > m, so no horizontal asmtote. Slant asmtote: g(x) = x + x = x x 6. a. C(x) =, + x 9

16 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr b. x +, C = x ( ) +, c. C( ) = = When calculators are manufactured, it costs $ to manufacture each. ( ) +, C( ) = = When calculators are manufactured, it costs $ to manufacture each. ( ) +, C( ) = = 7 When, calculators are manufactured, it costs $7 to manufacture each. (, ) +, C(, ) = =., When, calculators are manufactured, it costs $. to manufacture each. d. n = m, so = = is the horizontal asmtote. Minimum costs will aroach $. 66. a. C( 9) C( ) ( 9 ) ( ) 9 C(9) C() = 8 C(9) C() = 6 The difference in cost of removing 9% versus % of the contaminants is 6 million dollars. 67. x + f ( x) =.x + n = m, so =. = The number of fish available in the ond aroaches,,. 68. Px ( )= 7, 9 x + 79 n < m so As the number of ears of education increases the ercentage rate of unemloment aroaches zero..96x a. q( x) =.x b. =. 6. The ercentage of inmates that are in for violent crimes will aroach 6.%. c. Answers ma var. 7. b = e 98 = 7. b 7. e b. 7( ) = $ 7. d = t = () = 6 d = 6t d = 6( ) =, 6 ft b. x = ; No amount of mone can remove % of the contaminants, since C(x) increases without bound as x aroaches.

17 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions 7. t = r = t = r t = = hours 7. l = d 8 = 8 = l = d 79 l = = decibels c 7. t = w = 6 = h h = c t = w ( ) t = = 6 hours 7. V = hb 7 = = V = hb V = = 8 ft ( x + ) + x + x+ = x = ± = ± i no x-intercets -intercet: f ( ) = ( + ) + = 8 (, ) (, ) x. f( x)= x x b x = = = a f () = () = vertex: (, ) axis of smmetr x = x-intercets: x x = ( x )( x+ ) x = or x = -intercet: f ( ) ( ) = (, ) (, ) (, ) (, ) x Chater Test. f( x)= ( x+ ) + vertex: (, ) axis of smmetr: x = x-intercets:. f ( x) = x + x 6 Since the coefficient of x is negative, the grah of f(x) oens down and f(x) has a

18 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr maximum oint. x = = ( ) f () = () + () 6 = = maximum oint: (, ). f () x = x + 6x 6 x = a b = 6 = VCRs will maximize rofit. f ( )= ( ) + 6( ) 6 = 69 Maximum dail rofit = $6,9.. a. f( x)= x x x+ x x x+ x ( x ) ( x ) ( x )( x )( x+ ) x =,, The solution set is {,, }. b. The degree of the olnomial is odd and the leading coefficient is ositive. Thus the grah falls to the left and rises to the right. x 7. a. The integral root aears to be. b x 7x + (x )(x ) x = or x = The other two roots are and. 8. x + x 7x 6 : ±, ±, ±, ±6 q: ±, ± q : ±, ±, ±, ± 6, ±, ± 9. f( x)= x x x + x f(x) has sign variations. f ( x)= x x x x f( x) has no sign variations. There are or ositive real solutions and no negative real solutions.. x + 6x x : ±, ±, ±, ±, ±6, ±, ±, ± q: ± : ±, ±, ±, ±, ± 6, ±, ±, ± q 6 6. f( x)= x x Since the degree of the olnomial is odd and the leading coefficient is ositive, the grah of f should fall to the left and rise to the right. The x-intercets should be and. 6 x + 6x x ( x+ )( x + x 6) ( x+ )( x+ )( x ) x = or x = or x = The solution set is {,, }.

19 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions. f( x)= x x x + x+ a. : ±, ±, ±, ± q: ±, ± q : ±, ±, ±, ±, ±, ±, ±, ± b. ( x+ )( x x x+ ) ( x+ )[ x ( x ) ( x )] ( x+ )( x ) ( x )= x = or x = or x =± The solution set is,,,.. x + x 7x x = is a lower bound is an uer bound.. ( x + i) ( x i)= x x + x x + 6 x x + ) x 7x + 8x x + x x + x x + 6x x x + x x 6x x + 6x x + x x + 6 (x )(x ) x = or x = The solution set is {,, + i, i}.. f(x) has zeros at and. The zero at has multilicit of. x + x = (x )(x + ) x x 6 domain: {x x, x }. f( x)= Smmetr: f ( x) = -axis smmetr x-intercet: x -intercet: = x x 6 = f (x) 6 Vertical asmtotes: x 6 (x )(x + ) x =, Horizontal asmtote: n < m, so is the horizontal asmtote. x

20 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr 6. f( x)= x 9 x domain: {x x } Smmetr: f ( x) = x 9 x f( x) f(x), f( x) f(x) No smmetr x-intercets: x 9 (x )(x + ) x =, -intercet: 9 = 9 Vertical asmtote: x x = Horizontal asmtote: n > m, so no horizontal asmtote exists. Slant asmtote: f (x) = x + x = x + x 7. f( x)= x + x + x x + x = ( x+ )( x ) domain: {x x =, x } x + Smmetr: f ( x) = x x f( x) f(x), f( x) f(x) No smmetr x-intercet: x + x = -intercet: = + + ( ) = Vertical asmtotes: x + x (x + )(x ) x, Horizontal asmtote: n < m, so is the horizontal asmtote. x 8. f( x)= x x + domain: all real numbers Smmetr: f ( x) = -axis smmetr x-intercet: x x -intercet: = ( ) + Vertical asmtote: x + x = No vertical asmtote. Horizontal asmtote: x x + = f (x) n = m, so = = is the horizontal asmtote. x

21 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions 9. a. When x =, =.9 After learning tries, 9% of the resonses were correct. b. When x =, =.9 After learning tries, 9% of the resonses were correct. c. =.9/.9 = As the number of learning tries increases, the correct resonses aroaches %.. I = d = d = I = d I = = foot-candles Cumulative Review Exercises (Chaters P ). x = x = x = x = x = x = x = The solution set is {, }. 6. x x+ = x = ± = ± The solution set is, = x x 9x + x = 9x + x ( x )( x+ ) x x+ x = or x = The solution set is, = + = +. ( x x+ ) x + x = x 9x+ 6x x+ 8 = x x+ ( ). 8 + = + 6 = + = = ( ) 7. x x = x x = x ( x )( x+ )

22 Chater : Polnomial and Rational Functions SSM: Algebra and Trigonometr 8. x + x x 6 : ±, ±, ±, ±6 q: ± : ±, ±, ±, ± 6 q 6 6 radius: x x + x x 6 ( x+ )( x x ) ( x+ )( x+ )( x ) x = or x = or x = The solution set is {,, }. 9. x > x > x > 8 x > x < x < x < (, ) or (, ). x > x+ x x > x x = ( x )( x+ ) x = or x = Test intervals are (, ),,,,. Testing oints, the solution is (, ) or,.. x + x+ x x = + + ( x ) + ( + ) = 9 center: (, ). V = C( t) V = t C V C = t V t = C. f( x)= 9x 9x 9x x Domain: (,]. ( f g)( x) = x + x ( x ) = x + x x+ = x x. ( f o g)( x) = ( x ) + ( x ) = 6x 8x+ + 8x = 6x 6 6. gf ( ( )) f ( ) = ( ) + ( ) = 9 6 = g( ) = ( ) = 8 = 9 7. f( x)= x x x + 6

23 SSM: Algebra and Trigonometr Chater : Polnomial and Rational Functions a. x x x+ x ( x ) ( x )= ( x ) ( x )= ( x )( x+ )( x ) x =,, The solution set is {,, }. b. The grah falls to the left and rises to the right. x n =, a n so the grah falls to the left and rises to the right. x x. f( x)= x vertical asmtote: x = horizontal asmtote: = x-intercet: x = 8. f( x) = x + x 8 b x = = = a f ( ) = ( ) + ( ) 8 = 8= 9 vertex: (, 9) x-intercets: x + x 8= ( x+ )( x ) x = or x = -intercet: f() = 8 -intercet: = x x 9. f( x) = x ( x ) zeros: x (multilicit ) and x = -intercet: f( x)= x x 7