Checkpoint
f(1)=3f(1)=3 and f(a+h)=a2+2ah+h2−3a−3h+5f(a+h)=a2+2ah+h2−3a−3h+5
Domain = {x|x≤2},{x|x≤2}, range = {y|y≥5}{y|y≥5}
x = 0 , 2 , 3 x = 0 , 2 , 3
(fg)(x)=x2+32x−5.(fg)(x)=x2+32x−5. The domain is {x|x≠52}.{x|x≠52}.
( f ∘ g ) ( x ) = 2 − 5 x . ( f ∘ g ) ( x ) = 2 − 5 x .
( g ∘ f ) ( x ) = 0.63 x ( g ∘ f ) ( x ) = 0.63 x
f(x)f(x) is odd.
Domain = (−∞,∞),(−∞,∞), range = {y|y≥−4}.{y|y≥−4}.
m=1/2.m=1/2. The point-slope form is
y − 4 = 1 2 ( x − 1 ) . y − 4 = 1 2 ( x − 1 ) .
The slope-intercept form is
y = 1 2 x + 7 2 . y = 1 2 x + 7 2 .
The zeros are x=1±3/3.x=1±3/3. The parabola opens upward.
The domain is the set of real numbers xx such that x≠1/2.x≠1/2. The range is the set {y|y≠5/2}.{y|y≠5/2}.
The domain of ff is (−∞, ∞).(−∞, ∞). The domain of gg is {x|x≥1/5}.{x|x≥1/5}.
Algebraic
C ( x ) = { 49 , 0 < x ≤ 1 70 , 1 < x ≤ 2 91 , 2 < x ≤ 3 C ( x ) = { 49 , 0 < x ≤ 1 70 , 1 < x ≤ 2 91 , 2 < x ≤ 3
Shift the graph y=x2y=x2 to the left 1 unit, reflect about the xx-axis, then shift down 4 units.
7π/6;7π/6; 330°
cos ( 3 π / 4 ) = − 2 / 2 ; sin ( − π / 6 ) = −1 / 2 cos ( 3 π / 4 ) = − 2 / 2 ; sin ( − π / 6 ) = −1 / 2
1010 ft
θ=3π2+2nπ,π6+2nπ,5π6+2nπθ=3π2+2nπ,π6+2nπ,5π6+2nπ for n=0,±1,±2,…n=0,±1,±2,…
1+θ=1+θθ=θθ+θθ=θ+θθ=1θ=θ1+θ=1+θθ=θθ+θθ=θ+θθ=1θ=θ
To graph f(x)=3sin(4x)−5,f(x)=3sin(4x)−5, the graph of y=sin(x)y=sin(x) needs to be compressed horizontally by a factor of 4, then stretched vertically by a factor of 3, then shifted down 5 units. The function ff will have a period of π/2π/2 and an amplitude of 3.
No.
f−1(x)=2xx−3.f−1(x)=2xx−3. The domain of f−1f−1 is {x|x≠3}.{x|x≠3}. The range of f−1f−1 is {y|y≠2}.{y|y≠2}.
The domain of f−1f−1 is (0,∞).(0,∞). The range of f−1f−1 is (−∞,0).(−∞,0). The inverse function is given by the formula f−1(x)=−1/x.f−1(x)=−1/x.
f ( 4 ) = 900 ; f ( 10 ) = 24 , 300 . f ( 4 ) = 900 ; f ( 10 ) = 24 , 300 .
x / ( 2 y 3 ) x / ( 2 y 3 )
A(t)=750e0.04t.A(t)=750e0.04t. After 3030 years, there will be approximately $2,490.09.$2,490.09.
x = ln 3 2 x = ln 3 2
x = 1 e x = 1 e
1.29248 1.29248
The magnitude 8.48.4 earthquake is roughly 1010 times as severe as the magnitude 7.47.4 earthquake.
( x 2 + x −2 ) / 2 ( x 2 + x −2 ) / 2
1 2 ln ( 3 ) ≈ 0.5493 . 1 2 ln ( 3 ) ≈ 0.5493 .
Section 1.1 Exercises
a. Domain = {−3,−2,−1,0,1,2,3},{−3,−2,−1,0,1,2,3}, range = {0,1,4,9}{0,1,4,9} b. Yes, a function
a. Domain = {0,1,2,3},{0,1,2,3}, range = {−3,−2,−1,0,1,2,3}{−3,−2,−1,0,1,2,3} b. No, not a function
a. Domain = {3,5,8,10,15,21,33},{3,5,8,10,15,21,33}, range = {0,1,2,3}{0,1,2,3} b. Yes, a function
a. −2−2 b. 3 c. 13 d. −5x−2−5x−2 e. 5a−25a−2 f. 5a+5h−25a+5h−2
a. Undefined b. 2 c. 2323 d. −2x−2x e 2a2a f. 2a+h2a+h
a. 55 b. 1111 c. 2323 d. −6x+5−6x+5 e. 6a+56a+5 f. 6a+6h+56a+6h+5
a. 9 b. 9 c. 9 d. 9 e. 9 f. 9
x≥18;y≥0;x=18;x≥18;y≥0;x=18; no y-intercept
x ≥ −2 ; y ≥ −1 ; x = −1 ; y = −1 + 2 x ≥ −2 ; y ≥ −1 ; x = −1 ; y = −1 + 2
x≠4;y≠0;x≠4;y≠0; no x-intercept; y=−34y=−34
x>5;y>0;x>5;y>0; no intercepts
Function; a. Domain: all real numbers, range: y≥0y≥0 b. x=±1x=±1 c. y=1y=1 d. −1<x<0−1<x<0 and 1<x<∞1<x<∞ e. −∞<x<−1−∞<x<−1 and 0<x<10<x<1 f. Not constant g. y-axis h. Even
Function; a. Domain: all real numbers, range: −1.5≤y≤1.5−1.5≤y≤1.5 b. x=0x=0 c. y=0y=0 d. all real numbersall real numbers e. None f. Not constant g. Origin h. Odd
Function; a. Domain: −∞<x<∞,−∞<x<∞, range: −2≤y≤2−2≤y≤2 b. x=0x=0 c. y=0y=0 d. −2<x<2−2<x<2 e. Not decreasing f. −∞<x<−2−∞<x<−2 and 2<x<∞2<x<∞ g. Origin h. Odd
Function; a. Domain: −4≤x≤4,−4≤x≤4, range: −4≤y≤4−4≤y≤4 b. x=1.2x=1.2 c. y=4y=4 d. Not increasing e. 0<x<40<x<4 f. −4<x<0−4<x<0 g. No Symmetry h. Neither
a. 5x2+x−8;5x2+x−8; all real numbers b. −5x2+x−8;−5x2+x−8; all real numbers c. 5x3−40x2;5x3−40x2; all real numbers d. x−85x2;x≠0x−85x2;x≠0
a. −2x+6;−2x+6; all real numbers b. −2x2+2x+12;−2x2+2x+12; all real numbers c. −x4+2x3+12x2−18x−27;−x4+2x3+12x2−18x−27; all real numbers d. −x+3x+1;x≠−1,3−x+3x+1;x≠−1,3
a. 6+2x;x≠06+2x;x≠0 b. 6; x≠0x≠0 c. 6x+1x2;x≠06x+1x2;x≠0 d. 6x+1;x≠06x+1;x≠0
a. 4x+3;4x+3; all real numbers b. 4x+15;4x+15; all real numbers
a. x4−6x2+16;x4−6x2+16; all real numbers b. x4+14x2+46;x4+14x2+46; all real numbers
a. 3x4+x;x≠0,−43x4+x;x≠0,−4 b. 4x+23;x≠−124x+23;x≠−12
a. Yes, because there is only one winner for each year. b. No, because there are three teams that won more than once during the years 2001 to 2012.
a. V(s)=s3V(s)=s3 b. V(11.8)≈1643;V(11.8)≈1643; a cube of side length 11.8 each has a volume of approximately 1643 cubic units.
a. N(x)=15xN(x)=15x b. i. N(20)=15(20)=300;N(20)=15(20)=300; therefore, the vehicle can travel 300 mi on a full tank of gas. Ii. N(15)=225;N(15)=225; therefore, the vehicle can travel 225 mi on 3/4 of a tank of gas. c. Domain: 0≤x≤20;0≤x≤20; range: [0,300][0,300] d. The driver had to stop at least once, given that it takes approximately 39 gal of gas to drive a total of 578 mi.
a. A(t)=A(r(t))=π·(6−5t2+1)2A(t)=A(r(t))=π·(6−5t2+1)2 b. Exact: 121π4;121π4; approximately 95 cm2 c. C(t)=C(r(t))=2π(6−5t2+1)C(t)=C(r(t))=2π(6−5t2+1) d. Exact: 11π;11π; approximately 35 cm
a. S(x)=8.5x+750S(x)=8.5x+750 b. $962.50, $1090, $1217.50 c. 77 skateboards
Section 1.2 Exercises
a. −1 b. Decreasing
a. 3/4 b. Increasing
a. 4/3 b. Increasing
a. 0 b. Horizontal
y = −6 x + 9 y = −6 x + 9
y = 1 3 x + 4 y = 1 3 x + 4
y = 1 2 x y = 1 2 x
y = 3 5 x − 3
a. (m=2,b=−3) b.
a. (m=−6,b=0) b.
a. (m=0,b=−6) b.
a. (m=−23,b=2) b.
a. 2 b. 52,−1 c. −5 d. ਜਿਵੇਂ x → ±∞, y → ∞ e. ਕੋਈ ਨਹੀਂ
a. 2 b. ±2 c. −1 d. ਜਿਵੇਂ x → ±∞, y → ∞ e. ਸਮ
a. 3 b. 0, ±3 c. 0 d. ਜਿਵੇਂ x → -∞, y → ∞ ਅਤੇ x → ∞, y → -∞ e. ਟਾਂਕ
a. 13,−3,5 b.
a. −32,−12,4 b.
ਸੱਚ; n=3
ਝੂਠ; f(x)=xb, ਜਿੱਥੇ b ਇੱਕ ਅਸਲ-ਮੁੱਲ ਵਾਲਾ ਸਥਿਰ ਅੰਕ ਹੈ, ਇੱਕ ਘਾਤ ਫੰਕਸ਼ਨ ਹੈ
a. V(t)=−2733t+20500 b. (0,20,500) ਦਾ ਮਤਲਬ ਹੈ ਕਿ ਉਪਕਰਨ ਦੀ ਸ਼ੁਰੂਆਤੀ ਖਰੀਦ ਕੀਮਤ $20,500 ਹੈ; (7.5,0) ਦਾ ਮਤਲਬ ਹੈ ਕਿ 7.5 ਸਾਲਾਂ ਵਿੱਚ ਕੰਪਿਊਟਰ ਉਪਕਰਨ ਦਾ ਕੋਈ ਮੁੱਲ ਨਹੀਂ ਹੈ। c. $6835 d. ਲਗਭਗ 6.4 ਸਾਲਾਂ ਵਿੱਚ
a. C=0.75x+125 b. $245 c. 167 ਕੱਪਕੇਕ
a. V(t)=−1500t+26,000 b. 4 ਸਾਲਾਂ ਵਿੱਚ, ਕਾਰ ਦਾ ਮੁੱਲ $20,000 ਹੈ।
$30,337.50
ਕੁੱਲ ਸਮਰੱਥਾ ਦਾ 96%
ਸੈਕਸ਼ਨ 1.3 ਅਭਿਆਸ
4 π 3 ਰੇਡ
− π 3
11 π 6 ਰੇਡ
210 °
−540 °
−0.5
− 2 2 − 2 2
3 − 1 2 2 3 − 1 2 2
a. b=5.7b=5.7 b. sinA=47,cosA=5.77,tanA=45.7,cscA=74,secA=75.7,cotA=5.74sinA=47,cosA=5.77,tanA=45.7,cscA=74,secA=75.7,cotA=5.74
a. c=151.7c=151.7 b. sinA=0.5623,cosA=0.8273,tanA=0.6797,cscA=1.778,secA=1.209,cotA=1.471sinA=0.5623,cosA=0.8273,tanA=0.6797,cscA=1.778,secA=1.209,cotA=1.471
a. c=85c=85 b. sinA=8485,cosA=1385,tanA=8413,cscA=8584,secA=8513,cotA=1384sinA=8485,cosA=1385,tanA=8413,cscA=8584,secA=8513,cotA=1384
a. y=2425y=2425 b. sinθ=2425,cosθ=725,tanθ=247,cscθ=2524,secθ=257,cotθ=724sinθ=2425,cosθ=725,tanθ=247,cscθ=2524,secθ=257,cotθ=724
a. x=−23x=−23 b. sinθ=73,cosθ=−23,tanθ=−142,cscθ=377,secθ=−322,cotθ=−147sinθ=73,cosθ=−23,tanθ=−142,cscθ=377,secθ=−322,cotθ=−147
sec 2 x sec 2 x
sin 2 x sin 2 x
sec 2 θ sec 2 θ
1 sin t ( = csc t ) 1 sin t ( = csc t )
{ π 6 , 5 π 6 } { π 6 , 5 π 6 }
{ π 4 , 3 π 4 , 5 π 4 , 7 π 4 } { π 4 , 3 π 4 , 5 π 4 , 7 π 4 }
{ 2 π 3 , 5 π 3 } { 2 π 3 , 5 π 3 }
{ 0 , π , π 3 , 5 π 3 } { 0 , π , π 3 , 5 π 3 }
y = 4 sin ( π 4 x ) y = 4 sin ( π 4 x )
y = cos ( 2 π x ) y = cos ( 2 π x )
a. 1 b. 2π2π c. π4π4 units to the right
a. 1212 b. 8π8π c. No phase shift
a. 3 b. 22 c. 2π2π units to the left
Approximately 42 in.
a. 0.550 rad/sec b. 0.236 rad/sec c. 0.698 rad/min d. 1.697 rad/min
≈ 30.9 in 2 ≈ 30.9 in 2
a. π/184; the voltage repeats every π/184 sec b. Approximately 59 periods
a. Amplitude = 10;period=2410;period=24 b. 47.4°F47.4°F c. 14 hours later, or 2 p.m. d.
Section 1.4 Exercises
Not one-to-one
Not one-to-one
One-to-one
a. f−1(x)=x+4f−1(x)=x+4 b. Domain :x≥−4,range:y≥0:x≥−4,range:y≥0
a. f−1(x)=x−13f−1(x)=x−13 b. Domain: all real numbers, range: all real numbers
a. f−1(x)=x2+1,f−1(x)=x2+1, b. Domain: x≥0,x≥0, range: y≥1y≥1
These are inverses.
These are not inverses.
These are inverses.
These are inverses.
π 6 π 6
π 4 π 4
π 6 π 6
2 2 2 2
− π 6 − π 6
a. x=f−1(V)=0.04−V500x=f−1(V)=0.04−V500 b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity V. c. 0.1 cm; 0.14 cm; 0.17 cm
a. $31,250, $66,667, $107,143 b. (p=85CC+75)(p=85CC+75) c. 34 ppb
a. ~92°~92° b. ~42°~42° c. ~27°~27°
x≈6.69,8.51;x≈6.69,8.51; so, the temperature occurs on June 21 and August 15
~ 1.5 sec ~ 1.5 sec
tan−1(tan(2.1))≈−1.0416;tan−1(tan(2.1))≈−1.0416; the expression does not equal 2.1 since 2.1>1.57=π22.1>1.57=π2—in other words, it is not in the restricted domain of tanx.cos−1(cos(2.1))=2.1,tanx.cos−1(cos(2.1))=2.1, since 2.1 is in the restricted domain of cosx.cosx.
Section 1.5 Exercises
a. 125 b. 2.24 c. 9.74
a. 0.01 b. 10,000 c. 46.42
d
b
e
ਡੋਮੇਨ: ਸਾਰੇ ਵਾਸਤਵਿਕ ਅੰਕ, ਰੇਂਜ: (2,∞), y=2(2,∞), y=2
ਡੋਮੇਨ: ਸਾਰੇ ਵਾਸਤਵਿਕ ਅੰਕ, ਰੇਂਜ: (0,∞), y=0(0,∞), y=0
ਡੋਮੇਨ: ਸਾਰੇ ਵਾਸਤਵਿਕ ਅੰਕ, ਰੇਂਜ: (−∞,1), y=1(−∞,1), y=1
ਡੋਮੇਨ: ਸਾਰੇ ਵਾਸਤਵਿਕ ਅੰਕ, ਰੇਂਜ: (−1,∞), y=−1(−1,∞), y=−1
8 1 / 3 = 2 8 1 / 3 = 2
5 2 = 25 5 2 = 25
e −3 = 1 / e 3 e −3 = 1 / e 3
e 0 = 1 e 0 = 1
ਲੌਗ 4 ( 1 / 16 ) = −2 ਲੌਗ 4 ( 1 / 16 ) = −2
ਲੌਗ 9 1 = 0 ਲੌਗ 9 1 = 0
ਲੌਗ 64 4 = 1 / 3 ਲੌਗ 64 4 = 1 / 3
ਲੌਗ 9 150 = y ਲੌਗ 9 150 = y
ਲੌਗ 4 0.125 = − 3 / 2 ਲੌਗ 4 0.125 = − 3 / 2
ਡੋਮੇਨ: (1,∞),(1,∞), ਰੇਂਜ: (−∞,∞), x=1(−∞,∞), x=1
ਡੋਮੇਨ: (0,∞),(0,∞), ਰੇਂਜ: (−∞,∞), x=0(−∞,∞), x=0
ਡੋਮੇਨ: (−1,∞),(−1,∞), ਰੇਂਜ: (−∞,∞), x=−1(−∞,∞), x=−1
2 + 3 ਲੌਗ 3 a − ਲੌਗ 3 b 2 + 3 ਲੌਗ 3 a − ਲੌਗ 3 b
3 / 2 + 1 / 2 ਲੌਗ 5 x + 3 / 2 ਲੌਗ 5 y 3 / 2 + 1 / 2 ਲੌਗ 5 x + 3 / 2 ਲੌਗ 5 y
− 3 / 2 + ln 6 − 3 / 2 + ln 6
ln 15 3 ln 15 3
3 2 3 2
log 7.21 log 7.21
2 3 + log 11 3 log 7 2 3 + log 11 3 log 7
x = 1 25 x = 1 25
x = 4 x = 4
x = 3 x = 3
1 + 5 1 + 5
( log 82 log 7 ≈ 2.2646 ) ( log 82 log 7 ≈ 2.2646 )
( log 211 log 0.5 ≈ − 7.7211 ) ( log 211 log 0.5 ≈ − 7.7211 )
( log 0.452 log 0.2 ≈ 0.4934 ) ( log 0.452 log 0.2 ≈ 0.4934 )
~ 17 , 491 ~ 17 , 491
Approximately $131,653 is accumulated in 5 years.
i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid
a. ~333~333 million b. 94 years from 2013, or in 2107
a. k≈0.0578k≈0.0578 b. ≈92≈92 hours
The San Francisco earthquake was 103.4or≈2512103.4or≈2512 times more intense than the Japanese earthquake.
Review Exercises
False
False
Domain: x>5,x>5, range: all real numbers
Domain: x>2x>2 and x<−4,x<−4, range: all real numbers
Degree of 3, yy-intercept: 0, zeros: 0, 3−1,−1−33−1,−1−3
cos2x-sin2x=cos2x=1-2sin2x=2cos2x-1cos2x-sin2x=cos2x=1-2sin2x=2cos2x-1
0 , ± 2 π 0 , ± 2 π
4
One-to-one; yes, the function has an inverse; inverse: f−1(x)=1yf−1(x)=1y
x ≥ − 3 2 , f −1 ( x ) = − 3 2 + 1 2 4 y − 7 x ≥ − 3 2 , f −1 ( x ) = − 3 2 + 1 2 4 y − 7
a. C(x)=300+7xC(x)=300+7x b. 100 shirts
The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2
78.51%