**Unformatted text preview: **1 FUNCTIONS AND MODELS
- The Download Monster 1.1 Four Ways to Represent a Function In exercises requiring estimations or approximations, your answers may vary slightly from the answers given here.
1. (a) The point (−1, −2) is on the graph of f , so f (−1) = −2. (b) When x = 2, y is about 2.8, so f (2) ≈ 2.8.
(c) f (x) = 2 is equivalent to y = 2. When y = 2, we have x = −3 and x = 1.
(d) Reasonable estimates for x when y = 0 are x = −2.5 and x = 0.3.
(e) The domain of f consists of all x-values on the graph of f . For this function, the domain is −3 ≤ x ≤ 3, or [−3, 3].
The range of f consists of all y-values on the graph of f . For this function, the range is −2 ≤ y ≤ 3, or [−2, 3]. (f ) As x increases from −1 to 3, y increases from −2 to 3. Thus, f is increasing on the interval [−1, 3].
3. From Figure 1 in the text, the lowest point occurs at about (t, a) = (12, −85). The highest point occurs at about (17, 115). Thus, the range of the vertical ground acceleration is −85 ≤ a ≤ 115. Written in interval notation, we get [−85, 115]. 5. No, the curve is not the graph of a function because a vertical line intersects the curve more than once. Hence, the curve fails the Vertical Line Test.
7. Yes, the curve is the graph of a function because it passes the Vertical Line Test. The domain is [−3, 2] and the range is [−3, −2) ∪ [−1, 3].
9. The person’s weight increased to about 160 pounds at age 20 and stayed fairly steady for 10 years. The person’s weight dropped to about 120 pounds for the next 5 years, then increased rapidly to about 170 pounds. The next 30 years saw a gradual
increase to 190 pounds. Possible reasons for the drop in weight at 30 years of age: diet, exercise, health problems.
11. The water will cool down almost to freezing as the ice 13. Of course, this graph depends strongly on the melts. Then, when the ice has melted, the water will geographical location! slowly warm up to room temperature. 15. As the price increases, the amount sold decreases. 17. 9 10 ¤ - The Download Monster CHAPTER 1 FUNCTIONS AND MODELS 19. (a) (b) From the graph, we estimate the number of
cell-phone subscribers worldwide to be about
92 million in 1995 and 485 million in 1999. 21. f (x) = 3x2 − x + 2. f (2) = 3(2)2 − 2 + 2 = 12 − 2 + 2 = 12.
f (−2) = 3(−2)2 − (−2) + 2 = 12 + 2 + 2 = 16.
f (a) = 3a2 − a + 2.
f (−a) = 3(−a)2 − (−a) + 2 = 3a2 + a + 2.
f (a + 1) = 3(a + 1)2 − (a + 1) + 2 = 3(a2 + 2a + 1) − a − 1 + 2 = 3a2 + 6a + 3 − a + 1 = 3a2 + 5a + 4.
2f (a) = 2 · f (a) = 2(3a2 − a + 2) = 6a2 − 2a + 4.
f (2a) = 3(2a)2 − (2a) + 2 = 3(4a2 ) − 2a + 2 = 12a2 − 2a + 2.
f (a2 ) = 3(a2 )2 − (a2 ) + 2 = 3(a4 ) − a2 + 2 = 3a4 − a2 + 2.
2
[f (a)]2 = 3a2 − a + 2 = 3a2 − a + 2 3a2 − a + 2 = 9a4 − 3a3 + 6a2 − 3a3 + a2 − 2a + 6a2 − 2a + 4 = 9a4 − 6a3 + 13a2 − 4a + 4. f (a + h) = 3(a + h)2 − (a + h) + 2 = 3(a2 + 2ah + h2 ) − a − h + 2 = 3a2 + 6ah + 3h2 − a − h + 2.
23. f (x) = 4 + 3x − x2 , so f (3 + h) = 4 + 3(3 + h) − (3 + h)2 = 4 + 9 + 3h − (9 + 6h + h2 ) = 4 − 3h − h2 , and (4 − 3h − h2 ) − 4
h(−3 − h)
f (3 + h) − f (3)
=
=
= −3 − h.
h
h
h 1
a−x
1
−
a−x
−1(x − a)
1
f (x) − f (a)
x
a
=
= xa =
=
=−
25.
x−a
x−a
x−a
xa(x − a)
xa(x − a)
ax
27. f (x) = x/(3x − 1) is defined for all x except when 0 = 3x − 1
is x ∈ R | x 6= 13 = −∞, 13 ∪ 13 , ∞ . ⇔ x = 13 , so the domain √
√
√
t + 3 t is defined when t ≥ 0. These values of t give real number results for t, whereas any value of t gives a real
√
number result for 3 t. The domain is [0, ∞). 29. f (t) = 31. h(x) = 1 √
4
x2 − 5x is defined when x2 − 5x > 0 ⇔ x(x − 5) > 0. Note that x2 − 5x 6= 0 since that would result in division by zero. The expression x(x − 5) is positive if x < 0 or x > 5. (See Appendix A for methods for solving
inequalities.) Thus, the domain is (−∞, 0) ∪ (5, ∞). - The Download Monster SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ¤ 33. f (x) = 5 is defined for all real numbers, so the domain is R, or (−∞, ∞). The graph of f is a horizontal line with y-intercept 5. 35. f (t) = t2 − 6t is defined for all real numbers, so the domain is R, or (−∞, ∞). The graph of f is a parabola opening upward since the coefficient
of t2 is positive. To find the t-intercepts, let y = 0 and solve for t. 0 = t2 − 6t = t(t − 6) ⇒ t = 0 and t = 6. The t-coordinate of the vertex is halfway between the t-intercepts, that is, at t = 3. Since f (3) = 32 − 6 · 3 = −9, the vertex is (3, −9).
√
x − 5 is defined when x − 5 ≥ 0 or x ≥ 5, so the domain is [5, ∞).
√
Since y = x − 5 ⇒ y 2 = x − 5 ⇒ x = y 2 + 5, we see that g is the 37. g(x) = top half of a parabola. 39. G(x) = 3x + |x|
. Since |x| =
x ⎧ 3x + x
⎪
⎨
x
G(x) =
⎪
3x
−x
⎩
x x
−x if x ≥ 0
if x < 0 ⎧ 4x
⎪
⎨
x
=
⎪
2x
⎩
if x < 0
x
if x > 0 , we have if x > 0
=
if x < 0 4 if x > 0 2 if x < 0 Note that G is not defined for x = 0. The domain is (−∞, 0) ∪ (0, ∞).
41. f (x) = x+2 if x < 0 1−x if x ≥ 0 43. f (x) = The domain is R. x + 2 if x ≤ −1
x2 if x > −1 Note that for x = −1, both x + 2 and x2 are equal to 1.
The domain is R. 45. Recall that the slope m of a line between the two points (x1 , y1 ) and (x2 , y2 ) is m = connecting those two points is y − y1 = m(x − x1 ). The slope of this line segment is
y − (−3) = 52 (x − 1). The function is f (x) = 52 x − 11
,
2 1 ≤ x ≤ 5. y2 − y1
and an equation of the line
x2 − x1
7 − (−3)
5
= , so an equation is
5−1
2 11 12 - The Download Monster ¤ CHAPTER 1 FUNCTIONS AND MODELS 47. We need to solve the given equation for y. √
x + (y − 1)2 = 0 ⇔ (y − 1)2 = −x ⇔ y − 1 = ± −x ⇔ √
−x. The expression with the positive radical represents the top half of the parabola, and the one with the negative
√
radical represents the bottom half. Hence, we want f (x) = 1 − −x. Note that the domain is x ≤ 0.
y =1± 49. For 0 ≤ x ≤ 3, the graph is the line with slope −1 and y-intercept 3, that is, y = −x + 3. For 3 < x ≤ 5, the graph is the line with slope 2 passing through (3, 0); that is, y − 0 = 2(x − 3), or y = 2x − 6. So the function is
f (x) = −x + 3 if 0 ≤ x ≤ 3
2x − 6 if 3 < x ≤ 5 51. Let the length and width of the rectangle be L and W . Then the perimeter is 2L + 2W = 20 and the area is A = LW . Solving the first equation for W in terms of L gives W = 20 − 2L
= 10 − L. Thus, A(L) = L(10 − L) = 10L − L2 . Since
2 lengths are positive, the domain of A is 0 < L < 10. If we further restrict L to be larger than W , then 5 < L < 10 would be
the domain.
53. Let the length of a side of the equilateral triangle be x. Then by the Pythagorean Theorem, the height y of the triangle satisfies y2 + 2
1
2x = x2 , so that y 2 = x2 − 14 x2 = 34 x2 and y = A = 12 (base)(height), we obtain A(x) = 12 (x) √ 3
x
2 = √ 3
2 x. √ 3 2
x ,
4 Using the formula for the area A of a triangle, with domain x > 0. 55. Let each side of the base of the box have length x, and let the height of the box be h. Since the volume is 2, we know that 2 = hx2 , so that h = 2/x2 , and the surface area is S = x2 + 4xh. Thus, S(x) = x2 + 4x(2/x2 ) = x2 + (8/x), with
domain x > 0.
57. The height of the box is x and the length and width are L = 20 − 2x, W = 12 − 2x. Then V = LW x and so V (x) = (20 − 2x)(12 − 2x)(x) = 4(10 − x)(6 − x)(x) = 4x(60 − 16x + x2 ) = 4x3 − 64x2 + 240x.
The sides L, W , and x must be positive. Thus, L > 0 ⇔ 20 − 2x > 0 ⇔ x < 10;
W > 0 ⇔ 12 − 2x > 0 ⇔ x < 6; and x > 0. Combining these restrictions gives us the domain 0 < x < 6.
59. (a) (b) On $14,000, tax is assessed on $4000, and 10%($4000) = $400.
On $26,000, tax is assessed on $16,000, and
10%($10,000) + 15%($6000) = $1000 + $900 = $1900. (c) As in part (b), there is $1000 tax assessed on $20,000 of income, so
the graph of T is a line segment from (10,000, 0) to (20,000, 1000).
The tax on $30,000 is $2500, so the graph of T for x > 20,000 is
the ray with initial point (20,000, 1000) that passes through
(30,000, 2500). - The Download Monster SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS ¤ 13 61. f is an odd function because its graph is symmetric about the origin. g is an even function because its graph is symmetric with respect to the y-axis.
63. (a) Because an even function is symmetric with respect to the y-axis, and the point (5, 3) is on the graph of this even function, the point (−5, 3) must also be on its graph.
(b) Because an odd function is symmetric with respect to the origin, and the point (5, 3) is on the graph of this odd function,
the point (−5, −3) must also be on its graph.
65. f (x) = x
.
x2 + 1 f (−x) = 67. f (x) = −x
x
−x
= 2
=− 2
= −f(x).
(−x)2 + 1
x +1
x +1 −x
x
x
, so f(−x) =
=
.
x+1
−x + 1
x−1 Since this is neither f (x) nor −f (x), the function f is
neither even nor odd. So f is an odd function. 69. f (x) = 1 + 3x2 − x4 . f (−x) = 1 + 3(−x)2 − (−x)4 = 1 + 3x2 − x4 = f (x).
So f is an even function. 1.2 Mathematical Models: A Catalog of Essential Functions
1. (a) f (x) = (b) g(x) = √
5
x is a root function with n = 5.
√
1 − x2 is an algebraic function because it is a root of a polynomial. (c) h(x) = x9 + x4 is a polynomial of degree 9.
(d) r(x) = x2 + 1
is a rational function because it is a ratio of polynomials.
x3 + x (e) s(x) = tan 2x is a trigonometric function.
(f ) t(x) = log10 x is a logarithmic function.
3. We notice from the figure that g and h are even functions (symmetric with respect to the y-axis) and that f is an odd function (symmetric with respect to the origin). So (b) y = x5 must be f . Since g is flatter than h near the origin, we must have
(c) y = x8 matched with g and (a) y = x2 matched with h. 14 ¤ - The Download Monster CHAPTER 1 FUNCTIONS AND MODELS 5. (a) An equation for the family of linear functions with slope 2 is y = f (x) = 2x + b, where b is the y-intercept. (b) f (2) = 1 means that the point (2, 1) is on the graph of f . We can use the
point-slope form of a line to obtain an equation for the family of linear
functions through the point (2, 1). y − 1 = m(x − 2), which is equivalent
to y = mx + (1 − 2m) in slope-intercept form. (c) To belong to both families, an equation must have slope m = 2, so the equation in part (b), y = mx + (1 − 2m),
becomes y = 2x − 3. It is the only function that belongs to both families.
7. All members of the family of linear functions f(x) = c − x have graphs that are lines with slope −1. The y-intercept is c. 9. Since f (−1) = f (0) = f(2) = 0, f has zeros of −1, 0, and 2, so an equation for f is f (x) = a[x − (−1)](x − 0)(x − 2), or f (x) = ax(x + 1)(x − 2). Because f (1) = 6, we’ll substitute 1 for x and 6 for f (x).
6 = a(1)(2)(−1) ⇒ −2a = 6 ⇒ a = −3, so an equation for f is f (x) = −3x(x + 1)(x − 2).
11. (a) D = 200, so c = 0.0417D(a + 1) = 0.0417(200)(a + 1) = 8.34a + 8.34. The slope is 8.34, which represents the change in mg of the dosage for a child for each change of 1 year in age.
(b) For a newborn, a = 0, so c = 8.34 mg.
13. (a) (b) The slope of 9
5 means that F increases 9
5 degrees for each increase ◦ of 1 C. (Equivalently, F increases by 9 when C increases by 5
and F decreases by 9 when C decreases by 5.) The F -intercept of
32 is the Fahrenheit temperature corresponding to a Celsius
temperature of 0. - The Download Monster SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS 15. (a) Using N in place of x and T in place of y, we find the slope to be equation is T − 80 = 16 (N − 173) ⇔ T − 80 = 16 N −
(b) The slope of 1
6 173
6 ¤ 10
1
T2 − T1
80 − 70
=
= . So a linear
=
N2 − N1
173 − 113
60
6
⇔ T = 16 N + 307
6 307
6 = 51.16 . means that the temperature in Fahrenheit degrees increases one-sixth as rapidly as the number of cricket chirps per minute. Said differently, each increase of 6 cricket chirps per minute corresponds to an increase of 1◦ F.
(c) When N = 150, the temperature is given approximately by T = 16 (150) +
17. (a) We are given 307
6 = 76.16 ◦ F ≈ 76 ◦ F. 4.34
change in pressure
=
= 0.434. Using P for pressure and d for depth with the point
10 feet change in depth
10 (d, P ) = (0, 15), we have the slope-intercept form of the line, P = 0.434d + 15.
(b) When P = 100, then 100 = 0.434d + 15 ⇔ 0.434d = 85 ⇔ d = 85
0.434 ≈ 195.85 feet. Thus, the pressure is 100 lb/in2 at a depth of approximately 196 feet.
19. (a) The data appear to be periodic and a sine or cosine function would make the best model. A model of the form f (x) = a cos(bx) + c seems appropriate.
(b) The data appear to be decreasing in a linear fashion. A model of the form f (x) = mx + b seems appropriate.
Some values are given to many decimal places. These are the results given by several computer algebra systems — rounding is left to the reader. (b) Using the points (4000, 14.1) and (60,000, 8.2), we obtain 21. (a) 8.2 − 14.1
(x − 4000) or, equivalently,
60,000 − 4000
y ≈ −0.000105357x + 14.521429. y − 14.1 = A linear model does seem appropriate. (c) Using a computing device, we obtain the least squares regression line y = −0.0000997855x + 13.950764. The following commands and screens illustrate how to find the least squares regression line on a TI-83 Plus.
Enter the data into list one (L1) and list two (L2). Press Find the regession line and store it in Y1 . Press to enter the editor. . 15 16 ¤ - The Download Monster CHAPTER 1 FUNCTIONS AND MODELS Note from the last figure that the regression line has been stored in Y1 and that Plot1 has been turned on (Plot1 is
highlighted). You can turn on Plot1 from the Y= menu by placing the cursor on Plot1 and pressing
. pressing Now press or by to produce a graph of the data and the regression line. Note that choice 9 of the ZOOM menu automatically selects a window
that displays all of the data.
(d) When x = 25,000, y ≈ 11.456; or about 11.5 per 100 population.
(e) When x = 80,000, y ≈ 5.968; or about a 6% chance.
(f ) When x = 200,000, y is negative, so the model does not apply.
23. (a) (b) A linear model does seem appropriate. Using a computing device, we obtain the least squares
regression line y = 0.089119747x − 158.2403249, where x is the year and y is the height in feet. (c) When x = 2000, the model gives y ≈ 20.00 ft. Note that the actual winning height for the 2000 Olympics is less than the
winning height for 1996—so much for that prediction.
(d) When x = 2100, y ≈ 28.91 ft. This would be an increase of 9.49 ft from 1996 to 2100. Even though there was an increase
of 8.59 ft from 1900 to 1996, it is unlikely that a similar increase will occur over the next 100 years.
25. Using a computing device, we obtain the cubic
function y = ax3 + bx2 + cx + d with
a = 0.0012937, b = −7.06142, c = 12,823,
and d = −7,743,770. When x = 1925,
y ≈ 1914 (million). - The Download Monster SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ¤ 17 1.3 New Functions from Old Functions
1. (a) If the graph of f is shifted 3 units upward, its equation becomes y = f (x) + 3. (b) If the graph of f is shifted 3 units downward, its equation becomes y = f (x) − 3.
(c) If the graph of f is shifted 3 units to the right, its equation becomes y = f (x − 3).
(d) If the graph of f is shifted 3 units to the left, its equation becomes y = f (x + 3).
(e) If the graph of f is reflected about the x-axis, its equation becomes y = −f (x).
(f ) If the graph of f is reflected about the y-axis, its equation becomes y = f (−x).
(g) If the graph of f is stretched vertically by a factor of 3, its equation becomes y = 3f (x).
(h) If the graph of f is shrunk vertically by a factor of 3, its equation becomes y = 13 f (x).
3. (a) (graph 3) The graph of f is shifted 4 units to the right and has equation y = f (x − 4). (b) (graph 1) The graph of f is shifted 3 units upward and has equation y = f (x) + 3.
(c) (graph 4) The graph of f is shrunk vertically by a factor of 3 and has equation y = 13 f(x).
(d) (graph 5) The graph of f is shifted 4 units to the left and reflected about the x-axis. Its equation is y = −f (x + 4).
(e) (graph 2) The graph of f is shifted 6 units to the left and stretched vertically by a factor of 2. Its equation is
y = 2f (x + 6).
5. (a) To graph y = f (2x) we shrink the graph of f horizontally by a factor of 2. (b) To graph y = f 1
x
2 we stretch the graph of f horizontally by a factor of 2. The point (4, −1) on the graph of f corresponds to the
The point (4, −1) on the graph of f corresponds to the
point 1
2 point (2 · 4, −1) = (8, −1). · 4, −1 = (2, −1). (c) To graph y = f (−x) we reflect the graph of f about
the y-axis. The point (4, −1) on the graph of f corresponds to the
point (−1 · 4, −1) = (−4, −1). (d) To graph y = −f (−x) we reflect the graph of f about
the y-axis, then about the x-axis. The point (4, −1) on the graph of f corresponds to the
point (−1 · 4, −1 · −1) = (−4, 1). 18 ¤ - The Download Monster CHAPTER 1 FUNCTIONS AND MODELS 7. The graph of y = f (x) = √
3x − x2 has been shifted 4 units to the left, reflected about the x-axis, and shifted downward 1 unit. Thus, a function describing the graph is
y= −1 · f (x + 4) reflect
about x-axis shift
4 units left − 1
shift
1 unit left This function can be written as
y = −f (x + 4) − 1 = − 3(x + 4) − (x + 4)2 − 1 = − √
3x + 12 − (x2 + 8x + 16) − 1 = − −x2 − 5x − 4 − 1 9. y = −x3 : Start with the graph of y = x3 and reflect about the x-axis. Note: Reflecting about the y-axis gives the same result since substituting −x for x gives
us y = (−x)3 = −x3 . 11. y = (x + 1)2 : Start with the graph of y = x2 and shift 1 unit to the left. 13. y = 1 + 2 cos x: Start with the graph of y = cos x, stretch vertically by a factor of 2, and then shift 1 unit upward. 15. y = sin(x/2): Start with the graph of y = sin x and stretch horizontally by a factor of 2. - The Download Monster SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ¤ √
x + 3 : Start with the graph of
√
y = x and shift 3 units to the left. 17. y = 19. y = 2
1
2 (x + 8x) = 12 (x2 + 8x + 16) − 8 = 12 (x + 4)2 − 8: Start with the graph of y = x2 , compress vertically by a factor of 2, shift 4 units to the left, and then shift 8 units downward. 0 0 0 0 21. y = 2/(x + 1): Start with the graph of y = 1/x, shift 1 unit to the left, and then stretch vertically by a factor of 2. 23. y = |sin x|: Start with the graph of y = sin x and reflect all the parts of the graph below the x-axis about the x-axis. 25. This is just like the solution to Example 4 except the amplitude of the curve (the 30◦ N curve in Figure 9 on June 21) is 14 − 12 = 2. So the function is L(t) = 12 + 2 sin 2π
365 (t − 80) . March 31 is the 90th day of the year, so the model gives L(90) ≈ 12.34 h. The daylight time (5:51 AM to 6:18 PM) is 12 hours and 27 minutes, or 12.45 h. The model value differs
from the actual value by 12.45−12.34
12.45 ≈ 0.009, less than 1%. 27. (a) To obtain y = f (|x|), the portion of the graph of y = f (x) to the right of the y-axis is reflected about the y-axis. (b) y = sin |x| (c) y = |x| 19 20 ¤ - The Download Monster CHAPTER 1 FUNCTIONS AND MODELS 29. f (x) = x3 + 2x2 ; g(x) = 3x2 − 1. D = R for both f and g. (f + g)(x) = (x3 + 2x2 ) + (3x2 − 1) = x3 + 5x2 − 1, D = R.
(f − g)(x) = (x3 + 2x2 ) − (3x2 − 1) = x3 − x2 + 1, D = R.
(f g)(x) = (x3 + 2x2 )(3x2 − 1) = 3x5 + 6x4 − x3 − 2x2 , D = R.
f
x3 + 2x2
(x) =
, D=
g
3x2 − 1
31. f (x) = x2 − 1, D = R; 1
x | x 6= ± √
3 since 3x2 − 1 6= 0. g(x) = 2x + 1, D = R. (a) (f ◦ g)(x) = f (g(x)) = f(2x + 1) = (2x + 1)2 − 1 = (4x2 + 4x + 1) − 1 = 4x2 + 4x, D = R. (b) (g ◦ f)(x) = g(f (x)) = g(x2 − 1) = 2(x2 − 1) + 1 = (2x2 − 2) + 1 = 2x2 − 1, D = R.
(c) (f ◦ f )(x) = f(f (x)) = f (x2 − 1) = (x2 − 1)2 − 1 = (x4 − 2x2 + 1) − 1 = x4 − 2x2 , D = R.
(d) (g ◦ g)(x) = g(g(x)) = g(2x + 1) = 2(2x + 1) + 1 = (4x + 2) + 1 = 4x + 3, D = R. 33. f (x) = 1 − 3x; g(x) = cos x. D = R for both f and g, and hence for their composites. (a) (f ◦ g)(x) = f (g(x)) = f(cos x) = 1 − 3 cos x. (b) (g ◦ f)(x) = g(f (x)) = g(1 − 3x) = cos(1 − 3x).
(c) (f ◦ f )(x) = f(f (x)) = f (1 − 3x) = 1 − 3(1 − 3x) = 1 − 3 + 9x = 9x − 2.
(d) (g ◦ g)(x) = g(g(x)) = g(cos x) = cos(cos x) [Note that this is not cos x · cos x.]
x+1
1
, D = {x | x 6= 0}; g(x) =
, D = {x | x 6= −2}
x
x+2
x+1
x+1
x+1
1
x+2
=
=
+
+
(a) (f ◦ g)(x) = f (g(x)) = f
x+1
x+2
x+2
x+2
x+1
x+2 35. f (x) = x + = x2 + 2x + ...

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