CHAPTER Functions and Graphs SECTION., Page. x + x + x x x. x + x x x x x. ( x ) ( x ) x 6 x x x x x + x x 7. x + x + x + 6 8 x 8 6 x x. x x 6 x 6 x x x 8 x x 8 + x..x +..6.x. x 6 ( n + ) ( n ) n + n. ( n + ) ( n ) ( ) ( ) n + 6 n + 6 n n 7. ( x+ )( x ) (x+ )( x ) x + x x x 8 x 7 x 7..8 x+.( x).8x+ 8.x.x 8 x. x x a b ( ) ± x ± x + 6 c ( ) ()( ) () ± 6 ± 8 x + 8 x or 8 6 x. x + x a b c ± ()( ) x ± + x ± x
Chapter /Functions and Graphs. x + x + a b c ± ()() x () ± 6 8 x ± 8 ± x ± x. x x a b c ( ) ± x ± x ± x + 6 6 6 6 ( ) ()( ) () 7. x + x a b c ( ) ( ) ( ) ± ( ) x ± + 6 x x ± 6 x ± x ±. x + x + a b c ± x ± 8 x + x x or. x x + x x a ( ) ± x ± x ± x b + ( ) ()() () c. x x ( x+ )( x ) x + x or x x. 8y + 8y 7 (8y )( y + ) 8y y 8 or y + y 7. x 7x x(x 7) x or x 7 x 7. ( x ) [( x ) ][( x ) + ] x 8 x 8 or x x. x + < x < 8 x < 7. ( x + ) x + 7 x 6 x + 7 8 x 8. x+ > x+ 6 x > x < 6. 6x + 6x 8 x. (x ) > ( x ) x+ > x 8 x > 8 x <. x + 7x > xx ( + 7) >. x + 7x+ < ( x+ )( x+ ) < The product is positive. The critical values are and 7. The product is negative. The critical values are and. xx ( 7) ( x+ )( x + ) (, 7) (, ) (, )
Section.. x x 8 x x 8 ( x+ )( x 7) The product is positive or zero. The critical values are and 7. ( x+ )( x 7) 7. 6x x 6x x (x )(x+ ) The product is negative or zero. The critical values are and. (x )(x+ ) (, ] [7, ),. x < 6. x < 6. x + > < x < (, ) < x < 8 < x < ( 8, ) x + < x < or (, ) (7, ) x + > x > 7 6. x > 67. x + 6. x x < x < x < or x > x > x > x + x 8 or x + x (, 8] [, ) x x x 8,,, 8 7. x x x 8 x 8 or x x x 7. x Because an absolute value is always nonnegative, the inequality is always true. The solution set consists of all real numbers. (, ) 8 (, ], 7. x Because an absolute value is always nonnegative, the inequality x < has no solution. Thus, the only solution of the inequality x x. x is the solution of the equation
Chapter /Functions and Graphs 77. A A LW LW L W P 7 + W 7 W 7 + W 7 W W 7W + 7 P L+ W L+ W 7 (W 7)( W ) W 7 7 L 7 7L L or W LW L. L 7. A lw P 6 l + w l w l + w 6 + w 6 w, + w 6w w 6w+, ( w w+,) ( w )( w ) w ft l ft The dimensions are feet by feet. The rectangle measures. cm by cm. 8. Plan A: +.x Plan B: +.8x +.x < +.8x <.7x 7. < x Plan A is less expensive if you use at least 8 checks. 8. Plan A: + 8x Plan B: +.x + 8x > +.x.x > x >. Plan A pays better if at least sales are made. 8. 68 F 68 C + 6 C 7 C 87. nn ( + ) n + n 6 ( n+ )( n ) n So + + + L+ +. 8. R x x x x > x( x) > The product is positive. The critical values are and. x( x) (, )
Section.. s 6t + vt + s s > 8, v 6, s 6t + 6t > 8 6t + 6t 8 > 6( t t + ) > 6( t )( t ) > The product is positive. The critical values are t and t. 6( t )( t ). a. s.. b. s.. s.6 or s.. s. critical values:. and.6. s. 6 second < t < seconds The ball is higher than 8 ft between and seconds. SECTION., Page.. a. b. (8 6) + ( 7) + ( 87) + ( ) + (8 ) + ( 6) + ( 6) + (6 8) + ( 7) + ( 8) average + 7 + + + 8 + + + + + 6. The average increase in heart rate is. beats per minute.. d ( 8 6) + ( ) 7. d ( ( )) + ( ( )) ( ) + (7) ( 6) + () 6 + 6 + 6 7
Chapter /Functions and Graphs. d ( ) + ( ( 8)). d ( ) + ( 7 8) ( ) + (8) ( ) + ( ) + 6 8 ( ) + ( ) + (7 6 + 8) + 7 6 + 8 8 6. d ( a a) + ( b b). d ( x x) + (x x) with x< ( a) + ( b) ( x) + ( x) a + b x + x ( a + b ) x a + b x (Note: since x<, x x) 7. ( x) + (6 ) ( ) (6 ) x + 6 8x+ x + 6 x 8x 8 ( x )( x+ ) x or x The points are (, ), (, ). x. + x y, + y M + +, 6, (, ) 6 + 6 +. M, 8, (6, )..7 + (.). +.7 M,.7 7.8, (.87,.). 7.....
7.. Intercepts: ( ) ( ) Section.,, 6,. (, ), (, ), (, ) x y +. (, ), (, ), (, ). (, ± ), ( ±, ) 7. (, ± ), ( ±, ) x y x + y x + y. center (, ), radius 6. center (, ), radius. center (, ), radius 7. center (, ), radius 7. center (8, ), radius ψ. ( ) + ( y ) x + y x 6. ( ) 6. ( x ) + ( y ) r ( ) + ( ) r ( ) + r + 6 r r ( x ) + ( y ) 6. ( x+ ) + ( y ) r ( x ) + ( y ) r ( ) + ( ) r + ( ) r + 6 r r ( x ) + ( y ) 67. x 6 x + y x 6x+ + y + ( x ) + y center (, ), radius 6. x x + y y x x+ + y y+ + + ( x ) + ( y ) center (, ), radius 7. x x + y + 8 y 6 x x+ + y + 8y+ 6 6 + + 6 ( x 7) + ( y+ ) center (7, ), radius 7. x + x + y 6 6 x + x + y 6 x + x + + y + + + y 6 ( x ) ( x ) center ( ) + + ( y ),, radius 7. x x + y + y x x + + y + y + + x + y + 6 center,, radius 6 +
6 Chapter /Functions and Graphs 77. 7. 8. 8. 8. 87. x + y + 8., (, ) x + ( ) y + ( 8)., (, 7) therefore x + x + 8 x and y + y + 6 y x therefore x x 7 and y 8 7 y 8 y 6 Thus (, ) is the other endpoint. Thus (7, 6) is the other endpoint.. Let P(x, y) be the point three-fourths the distance from P ( 8, ) to P (6, ). d( P ) Then P(x, y) divides the line segment from P to P into the ratio to, or, P. d( P, P ) 8 + (6) 8 + 8 + () + 6 7 x and y + + Therefore r. 7 The coordinates of the point three-fourths the distance from P to P are,.. Let P(x, y) be the point seven-eights the distance from P (6, ) to P (, ). d( P ) Then P(x, y) divides the line segment from P to P into the ratio 7 to, or, P 7 7. Therefore r 7 d( P, P ) 6 + 7() 6 + + 7() + 7 7 x and y + 7 8 8 + 7 8 8 7 The coordinates of the point seven-eighths the distance from P to P are,. 8 7. d ((, ),(,)) ( ) + ( ( )) + 6 d((,),(,)) d((,),(, )) d((, ),(, )) ( ) + ( ) ( ( )) + ( ) ( ( )) + ( ( )) ( 6) + ( ) ( 6) 6 + ( ) The sides are all of equal length. Now we find the length of each diagonal: d((,),(, )) d((,),(, )) ( ) + ( ) ( ( )) + ( ) ( 8) + ( ) + ( 8) 8 Since the sides are of the same length and the diagonals are of equal length, we know these points are the vertices of a square. 8
Section. 7. ( x) + ( y) ( x) + ( y) 6x+ x + 6 8y+ y x 6x+ y 8y. ( x) + ( y) + ( x) + ( y) ( x) + ( y) ( x) + ( y) + ( x) + ( y ) 6 8x+ x + y ( x) + ( y) + 6 + 8x+ x + y 6x ( x) + ( y) x+ ( x) + ( y) 6x + x+ 6 ( x) ( y) + 6x + x+ 6 6 + 8x+ x + y 6x + x+ 6 + x+ x + y Simplifying yields x + y.. d ( ) + ( ) 6 + 6 Since the diameter is, the radius is. The center is the midpoint of the line segment from (, ) to (, ). + ( ) +, (,7) center ( x+ ) + ( y 7). Since it is tangent to the x-axis, its radius is. ( x 7) + ( y ) 7. The center is (,). The radius is. ( x + ) + ( y ) SECTION., Page 8. Given f ( x) x, a. f () () 6 b. f ( ) ( ) c. f () () d. f e. f ( k) ( k) k f. f ( k + ) ( k + ) k + 6 k +
8 Chapter /Functions and Graphs. Given A ( w ) w +, a. A () () + b. A () () + c. A ( ) ( ) + d. A () + e. A ( r + ) ( r + ) + r + r + + r + r + 6 f. A ( c ) ( c ) + + c. Given f ( x), x a. b. c. d. e. f () f ( ) f f() + f( ) + f( c + ) c + 7. Given s ( x) x, x a. s ( ) b. s ( ) c. s ( ) d. s ( ) e. Since t >, t t. s ( t) t t t t f. Since t <, t t. s ( t) t t t t f. c + f ( + h) + h. a. Since x <, use P( x) x +. P( ) ( ) + + b. Since x, use P ( x ) x +. ( ) ( ) + + 6 P c. Since x c <, use P( x) x +. P( c) c + d. Since k, then x k +, so use P ( x ) x +. P( k + ) ( k + ) + ( k + k + ) + k k + k k +. x + y 7 y x + 7 y x + 7, y is a function of x.. x+ y y x+ y ± x+, y is a not function of x.. y ± x, y is not a function of x since for each x > there are two values of x. 7. y x, y is a function of x.. y x y ± x, y is a not function of x.
Section.. Function; each x is paired with exactly one y.. Function; each x is paired with exactly one y.. Function; each x is paired with exactly one y. 7. f ( x) x Domain is the set of all real numbers.. f ( x ) x + Domain is the set of all real numbers.. f ( x) x + Domain is { x x }.. f ( x) 7 + x Domain is { x x 7}.. f ( x) x Domain is { x x }. 7. f ( x) x + Domain is { x x > }.. Domain: the set of all real numbers.. Domain: {,,,,,,,, } Domain: { x 6 x 6}. 7. a. C (.7)..int(.7)..int(.7)..( ). +.87 $.6 b. Domain: { x x }. a. Yes; every vertical line intersects the graph in one point. b. Yes; every vertical line intersects the graph in one point. c. No; some vertical lines intersect the graph at more than one point. d. Yes; every vertical line intersects the graph in at most one point.. Decreasing on (, ] ; increasing on [, ). Increasing on (, ). Decreasing on (, ] ; increasing on [, ] ; decreasing on [, ] ; increasing on [, )
Chapter /Functions and Graphs 7. Constant on (, ] ; increasing on [, ). Decreasing on (, ] ; constant on [, ]; increasing on [, ) 6. g and F are one-to-one since every horizontal line intersects the graph at one point. f, V, and p are not one-to-one since some horizontal lines intersect the graph at more than one point. 6. a. P l + w w l w l b. A lw A l( l) A l l 67. a. Cx ( ) () +.8x +.8x b. R( x) 7. x c. Px ( ) 7. x Cx ( ) 7. [ +.8 x] 7.x.8x.x 6. v ( t) 8, 6t, t 6. h r h r r h h r h( r) r Note x is a natural number. 7. d ( t) + () d t + meters, t 6 7. d ( 8t) + (6t) miles where t is the number of hours after : noon 7. x. Y(x) 7 7 8 answers accurate to nearest apple 77. f ( c) c c c c 6 ( c )( c + ) c or c + c c 7. is not in the range of f(x), since x only if x + x or. x + 8. Set the graphing utility to dot mode. WINDOW FORMAT Xmin-.7 Xmax.7 Xscl Ymin- Ymax Yscl 8. WINDOW FORMAT Xmin-.7 Xmax.7 Xscl Ymin- Ymax Yscl
Section. 8. 87. f ( x) ( ) ( ) 6 8. f ( x) (6 ). a. f (,7) () + (7) + 6 b. f (,) () + () c. f (,) ( ) + () d. f (,) () + () e. f ( k,k) ( k) + (k) k f. f( k +, k ) ( k + ) + ( k ) k + 6+ k 8k. a + a a a + a ( a )( a+ ) + 8 +. s A(,8,) ( )( 8)( ) (7)()() 6 7. a or a SECTION., Page... 7.... Replacing x by x leaves the equation unaltered. Thus the graph is symmetric with respect to the y-axis.. Not symmetric with respect to either axis. (neither) 7. Symmetric with respect to both the x- and the y-axes.. Symmetric with respect to both the x- and the y-axes.. Symmetric with respect to both the x- and the y-axes.. No, since ( y) ( x) simplifies to ( y) x, which is not equivalent to the original equation. Yes, since ( y) ( x) implies y x or y x, which is the original equation. 7. Yes, since ( x) + ( y) simplifies to the original equation. y x.. Yes, since x y simplifies to the original equation. x
. Chapter /Functions and Graphs.. symmetric with respect to the y-axis symmetric with respect to the origin symmetric with respect to the origin 7... symmetric with respect to the line x symmetric with respect to the line x no symmetry. Even, since g ( x) ( x) 7 x 7 g( x).. Odd, since F( x) ( x) + ( x) x x F( x). 7. Even. Even. Even. Even. Neither 7.. 6. y m( x) + 6. a. b.
Section. 6. a. b. 67. 6. 7. 7. 7. a. b. c. 77. a. f ( x) + ( x + ) + b. f ( x) ( x ) +