13.2.3: Chapter 3
( \newcommand{\kernel}{\mathrm{null}\,}\)
Try It
3.1 Complex Numbers
1.
−24−−−−√=0+2i6–√−24=0+2i6
2.
3.
(3−4i)−(2+5i)=1−9i(3−4i)−(2+5i)=1−9i
4.
−8−24i−8−24i
5.
18+i18+i
6.
102−29i102−29i
7.
−317+5i17−317+5i17
3.2 Quadratic Functions
1.
The path passes through the origin and has vertex at (−4,7),(−4,7), so (h)x=–716(x+4)2+7.(h)x=–716(x+4)2+7. To make the shot, h(−7.5)h(−7.5) would need to be about 4 but h(–7.5)≈1.64;h(–7.5)≈1.64; he doesn’t make it.
2.
g(x)=x2−6x+13g(x)=x2−6x+13 in general form; g(x)=(x−3)2+4g(x)=(x−3)2+4 in standard form
3.
The domain is all real numbers. The range is f(x)≥811,f(x)≥811, or [811,∞).[ 811,∞ ).
4.
y-intercept at (0, 13), No x-x- intercepts
5.
- ⓐ 3 seconds
- ⓑ 256 feet
- ⓒ 7 seconds
3.3 Power Functions and Polynomial Functions
1.
f(x)f(x) is a power function because it can be written as f(x)=8x5.f(x)=8x5. The other functions are not power functions.
2.
As xx approaches positive or negative infinity, f(x)f(x) decreases without bound: as x→±∞x→±∞, f(x)→−∞f(x)→−∞ because of the negative coefficient.
3.
The degree is 6. The leading term is −x6.−x6. The leading coefficient is −1.−1.
4.
As x→∞,f(x)→−∞;asx→−∞,f(x)→−∞.x→∞,f(x)→−∞;asx→−∞,f(x)→−∞. It has the shape of an even degree power function with a negative coefficient.
5.
The leading term is 0.2x3,0.2x3, so it is a degree 3 polynomial. As xx approaches positive infinity, f(x)f(x) increases without bound; as xx approaches negative infinity, f(x)f(x) decreases without bound.
6.
y-intercept (0,0);(0,0); x-intercepts (0,0),(–2,0),(0,0),(–2,0), and (5,0)(5,0)
7.
There are at most 12 x-x- intercepts and at most 11 turning points.
8.
The end behavior indicates an odd-degree polynomial function; there are 3 x-x- intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.
9.
The x-x- intercepts are (2,0),(−1,0),(2,0),(−1,0), and (5,0),(5,0), the y-intercept is (0,2),(0,2), and the graph has at most 2 turning points.
3.4 Graphs of Polynomial Functions
1.
y-intercept (0,0);(0,0); x-intercepts (0,0),(–5,0),(2,0),(0,0),(–5,0),(2,0), and (3,0)(3,0)
2.
The graph has a zero of –5 with multiplicity 3, a zero of –1 with multiplicity 2, and a zero of 3 with multiplicity 2.
3.
4.
Because ff is a polynomial function and since f(1)f(1) is negative and f(2)f(2) is positive, there is at least one real zero between x=1x=1 and x=2.x=2.
5.
f(x)=−18(x−2)3(x+1)2(x−4)f(x)=−18(x−2)3(x+1)2(x−4)
6.
The minimum occurs at approximately the point (0,−6.5),(0,−6.5), and the maximum occurs at approximately the point (3.5,7).(3.5,7).
3.5 Dividing Polynomials
1.
4x2−8x+15−784x+54x2−8x+15−784x+5
2.
3x3−3x2+21x−150+1,090x+73x3−3x2+21x−150+1,090x+7
3.
3x2−4x+13x2−4x+1
3.6 Zeros of Polynomial Functions
1.
f(−3)=−412f(−3)=−412
2.
The zeros are 2, –2, and –4.
3.
There are no rational zeros.
4.
The zeros are –4, 12, and 1.–4, 12, and 1.
5.
f(x)=−12x3+52x2−2x+10f(x)=−12x3+52x2−2x+10
6.
There must be 4, 2, or 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros.
7.
3 meters by 4 meters by 7 meters
3.7 Rational Functions
1.
End behavior: as x→±∞,f(x)→0;x→±∞,f(x)→0; Local behavior: as x→0,f(x)→∞x→0,f(x)→∞ (there are no x- or y-intercepts)
2.
The function and the asymptotes are shifted 3 units right and 4 units down. As x→3,f(x)→∞,x→3,f(x)→∞, and as x→±∞,f(x)→−4.x→±∞,f(x)→−4.
The function is f(x)=1(x−3)2−4.f(x)=1(x−3)2−4.
3.
12111211
4.
The domain is all real numbers except x=1x=1 and x=5.x=5.
5.
Removable discontinuity at x=5.x=5. Vertical asymptotes: x=0,x=1.x=0,x=1.
6.
Vertical asymptotes at x=2x=2 and x=–3;x=–3; horizontal asymptote at y=4.y=4.
7.
For the transformed reciprocal squared function, we find the rational form. f(x)=1(x−3)2−4=1−4(x−3)2(x−3)2=1−4(x2−6x+9)(x−3)(x−3)=−4x2+24x−35x2−6x+9f(x)=1(x−3)2−4=1−4(x−3)2(x−3)2=1−4(x2−6x+9)(x−3)(x−3)=−4x2+24x−35x2−6x+9
Because the numerator is the same degree as the denominator we know that as x→±∞,f(x)→−4;soy=–4x→±∞,f(x)→−4;soy=–4 is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is x=3,x=3, because as x→3,f(x)→∞.x→3,f(x)→∞. We then set the numerator equal to 0 and find the x-intercepts are at (2.5,0)(2.5,0) and (3.5,0).(3.5,0). Finally, we evaluate the function at 0 and find the y-intercept to be at (0,−359).(0,−359).
8.
Horizontal asymptote at y=12.y=12. Vertical asymptotes at x=1andx=3.x=1andx=3. y-intercept at (0,43.)(0,43.)
x-intercepts at (2,0) and (–2,0).(2,0) and (–2,0). (–2,0)(–2,0) is a zero with multiplicity 2, and the graph bounces off the x-axis at this point. (2,0)(2,0) is a single zero and the graph crosses the axis at this point.
3.8 Inverses and Radical Functions
1.
f−1(f(x))=f−1(x+53)=3(x+53)−5=(x−5)+5=xf−1(f(x))=f−1(x+53)=3(x+53)−5=(x−5)+5=x and f(f−1(x))=f(3x−5)=(3x−5)+53=3x3=xf(f−1(x))=f(3x−5)=(3x−5)+53=3x3=x
2.
f−1(x)=x3−4f−1(x)=x3−4
3.
f−1(x)=x−1−−−−√f−1(x)=x−1
4.
f−1(x)=x2−32,x≥0f−1(x)=x2−32,x≥0
5.
f−1(x)=2x+3x−1f−1(x)=2x+3x−1
3.1 Section Exercises
1.
Add the real parts together and the imaginary parts together.
3.
ii times ii equals –1, which is not imaginary. (answers vary)
5.
−8+2i−8+2i
7.
14+7i14+7i
9.
−2329+1529i−2329+1529i
11.
2 real and 0 nonreal
13.
15.
17.
8−i8−i
19.
−11+4i−11+4i
21.
2−5i2−5i
23.
6+15i6+15i
25.
−16+32i−16+32i
27.
−4−7i−4−7i
29.
25
31.
2−23i2−23i
33.
4−6i4−6i
35.
25+115i25+115i
37.
15i15i
39.
1+i3–√1+i3
41.
11
43.
−1−1
45.
128i
47.
(3√2+12i)6=−1(32+12i)6=−1
49.
3i3i
51.
0
53.
5 – 5i
55.
−2i−2i
57.
92−92i92−92i
3.2 Section Exercises
1.
When written in that form, the vertex can be easily identified.
3.
If a=0a=0 then the function becomes a linear function.
5.
If possible, we can use factoring. Otherwise, we can use the quadratic formula.
7.
g(x)=(x+1)2−4,g(x)=(x+1)2−4, Vertex (−1,−4)(−1,−4)
9.
f(x)=(x+52)2−334,f(x)=(x+52)2−334, Vertex (−52,−334)(−52,−334)
11.
f(x)=3(x−1)2−12,f(x)=3(x−1)2−12, Vertex (1,−12)(1,−12)
13.
f(x)=3(x−56)2−3712,f(x)=3(x−56)2−3712, Vertex (56,−3712)(56,−3712)
15.
Minimum is −172−172 and occurs at 52.52. Axis of symmetry is x=52.x=52.
17.
Minimum is −1716−1716 and occurs at −18.−18. Axis of symmetry is x=−18.x=−18.
19.
Minimum is −72−72 and occurs at −3.−3. Axis of symmetry is x=−3.x=−3.
21.
Domain is (−∞,∞).(−∞,∞). Range is [2,∞).[2,∞).
23.
Domain is (−∞,∞).(−∞,∞). Range is [−5,∞).[−5,∞).
25.
Domain is (−∞,∞).(−∞,∞). Range is [−12,∞).[−12,∞).
27.
{2i2–√,−2i2–√}{ 2i2,−2i2 }
29.
{3i3–√,−3i3–√}{ 3i3,−3i3 }
31.
{2+i,2−i}{2+i,2−i}
33.
{2+3i,2−3i}{2+3i,2−3i}
35.
{5+i,5−i}{5+i,5−i}
37.
{2+26–√,2−26–√}{2+26,2−26}
39.
{−12+32i,−12−32i}{ −12+32i,−12−32i }
41.
{−35+15i,−35−15i}{ −35+15i,−35−15i }
43.
{−12+12i7–√,−12−12i7–√}{ −12+12i7,−12−12i7 }
45.
f(x)=x2−4x+4f(x)=x2−4x+4
47.
f(x)=x2+1f(x)=x2+1
49.
f(x)=649x2+6049x+29749f(x)=649x2+6049x+29749
51.
f(x)=−x2+1f(x)=−x2+1
53.
Vertex (1,−1),(1,−1), Axis of symmetry is x=1.x=1. Intercepts are (0,0),(2,0).(0,0),(2,0).
55.
Vertex (52,−494),(52,−494), Axis of symmetry is x=52,x=52, intercepts: (6,0),(−1,0).(6,0), (−1,0).
57.
Vertex (54,−398),(54,−398), Axis of symmetry is x=54.x=54. Intercepts are (0,−8).(0,−8).
59.
f(x)=x2−4x+1f(x)=x2−4x+1
61.
f(x)=−2x2+8x−1f(x)=−2x2+8x−1
63.
f(x)=12x2−3x+72f(x)=12x2−3x+72
65.
f(x)=x2+1f(x)=x2+1
67.
f(x)=2−x2f(x)=2−x2
69.
f(x)=2x2f(x)=2x2
71.
The graph is shifted up or down (a vertical shift).
73.
50 feet
75.
Domain is (−∞,∞).(−∞,∞). Range is [−2,∞).[−2,∞).
77.
Domain is (−∞,∞)(−∞,∞) Range is (−∞,11].(−∞,11].
79.
f(x)=2x2−1f(x)=2x2−1
81.
f(x)=3x2−9f(x)=3x2−9
83.
f(x)=5x2−77f(x)=5x2−77
85.
50 feet by 50 feet. Maximize f(x)=−x2+100x.f(x)=−x2+100x.
87.
125 feet by 62.5 feet. Maximize f(x)=−2x2+250x.f(x)=−2x2+250x.
89.
66 and −6;−6; product is –36; maximize f(x)=x2+12x.f(x)=x2+12x.
91.
2909.56 meters
93.
$10.70
3.3 Section Exercises
1.
The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.
3.
As xx decreases without bound, so does f(x).f(x). As xx increases without bound, so does f(x).f(x).
5.
The polynomial function is of even degree and leading coefficient is negative.
7.
f(x)f(x) is a power function because it contains a variable base raised to a fixed power. It is also a polynomial, with all coefficients except one equal to zero.
9.
Neither
11.
Neither
13.
Degree = 2, Coefficient = –2
15.
Degree =4, Coefficient = –2
17.
As x→∞x→∞, f(x)→∞f(x)→∞, as x→−∞x→−∞, f(x)→∞f(x)→∞
19.
As x→−∞x→−∞, f(x)→−∞f(x)→−∞, as x→∞x→∞, f(x)→−∞f(x)→−∞
21.
As x→−∞x→−∞, f(x)→−∞f(x)→−∞,as x→∞x→∞, f(x)→−∞f(x)→−∞
23.
As x→∞x→∞, f(x)→∞f(x)→∞, as x→−∞x→−∞,f(x)→−∞f(x)→−∞
25.
y-intercept is (0,12),(0,12), t-intercepts are (1,0);(–2,0);and (3,0).(1,0);(–2,0);and (3,0).
27.
y-intercept is (0,−16).(0,−16). x-intercepts are (2,0)(2,0) and (−2,0).(−2,0).
29.
y-intercept is (0,0).(0,0). x-intercepts are (0,0),(4,0),(0,0),(4,0), and (−2,0).(−2,0).
31.
3
33.
5
35.
3
37.
5
39.
Yes. Number of turning points is 2. Least possible degree is 3.
41.
Yes. Number of turning points is 1. Least possible degree is 2.
43.
Yes. Number of turning points is 0. Least possible degree is 1.
44.
No.
45.
Yes. Number of turning points is 0. Least possible degree is 1.
47.
| xx | f(x)f(x) |
|---|---|
| 10 | 9,500 |
| 100 | 99,950,000 |
| –10 | 9,500 |
| –100 | 99,950,000 |
as x→−∞x→−∞, f(x)→∞f(x)→∞, as x→∞x→∞, f(x)→∞f(x)→∞
49.
| xx | f(x)f(x) |
|---|---|
| 10 | –504 |
| 100 | –941,094 |
| –10 | 1,716 |
| –100 | 1,061,106 |
as x→−∞x→−∞, f(x)→∞f(x)→∞, as x→∞x→∞, f(x)→−∞f(x)→−∞
51.
The y-y- intercept is (0,0).(0,0). The x-x- intercepts are (0,0),(2,0).(0,0),(2,0).As x→−∞x→−∞, f(x)→∞f(x)→∞, as x→∞x→∞, f(x)→∞f(x)→∞
53.
The y-y- intercept is (0,0)(0,0) . The x-x- intercepts are (0,0),(5,0),(7,0).(0,0),(5,0),(7,0). As x→−∞x→−∞, f(x)→∞f(x)→∞, as x→∞x→∞, f(x)→∞f(x)→∞
55.
The y-y- intercept is (0,0).(0,0). The x-x- intercept is (−4,0),(0,0),(4,0).(−4,0),(0,0),(4,0). As x→−∞x→−∞, f(x)→∞f(x)→∞, as x→∞x→∞, f(x)→∞f(x)→∞
57.
The y-y- intercept is (0,−81).(0,−81). The x-x- intercept are (3,0),(−3,0).(3,0),(−3,0). As x→−∞x→−∞, f(x)→∞f(x)→∞, as x→∞x→∞, f(x)→∞f(x)→∞
59.
The y-y- intercept is (0,0).(0,0). The x-x- intercepts are (−3,0),(0,0),(5,0).(−3,0),(0,0),(5,0). As x→−∞x→−∞, f(x)→∞f(x)→∞, as x→∞x→∞, f(x)→∞f(x)→∞
61.
f(x)=x2−4f(x)=x2−4
63.
f(x)=x3−4x2+4xf(x)=x3−4x2+4x
65.
f(x)=x4+1f(x)=x4+1
67.
V(m)=8m3+36m2+54m+27V(m)=8m3+36m2+54m+27
69.
V(x)=4x3−32x2+64xV(x)=4x3−32x2+64x
3.4 Section Exercises
1.
The x-x- intercept is where the graph of the function crosses the x-x- axis, and the zero of the function is the input value for which f(x)=0.f(x)=0.
3.
If we evaluate the function at aa and at bb and the sign of the function value changes, then we know a zero exists between aa and b.b.
5.
There will be a factor raised to an even power.
7.
(−2,0),(3,0),(−5,0)(−2,0),(3,0),(−5,0)
9.
(3,0),(−1,0),(0,0)(3,0),(−1,0),(0,0)
11.
(0,0),(−5,0),(2,0)(0,0),(−5,0),(2,0)
13.
(0,0),(−5,0),(4,0)(0,0),(−5,0),(4,0)
15.
(2,0),(−2,0),(−1,0)(2,0),(−2,0),(−1,0)
17.
(−2,0),(2,0),(12,0)(−2,0),(2,0),(12,0)
19.
(1,0),(−1,0)(1,0),(−1,0)
21.
(0,0),(3–√,0),(−3–√,0)(0,0),(3,0),(−3,0)
23.
(0,0)(0,0), (1,0)(1,0), (−1,0)(−1,0), (2,0)(2,0), (−2,0)(−2,0)
25.
f(2)=–10f(2)=–10 and f(4)=28.f(4)=28. Sign change confirms.
27.
f(1)=3f(1)=3 and f(3)=–77.f(3)=–77. Sign change confirms.
29.
f(0.01)=1.000001f(0.01)=1.000001 and f(0.1)=–7.999.f(0.1)=–7.999. Sign change confirms.
31.
0 with multiplicity 2, −32−32 with multiplicity 5, 4 with multiplicity 2
33.
0 with multiplicity 2, –2 with multiplicity 2
35.
−23−23 with multiplicity 5,55,5 with multiplicity 22
37.
00 with multiplicity 4,24,2 with multiplicity 1,–11,–1 with multiplicity 11
39.
3232 with multiplicity 2, 0 with multiplicity 3
41.
00 with multiplicity 6,236,23 with multiplicity 22
43.
x-intercepts, (1, 0)(1, 0) with multiplicity 2, (–4, 0)(–4, 0) with multiplicity 1, y-y- intercept (0, 4)(0, 4) . As x→−∞x→−∞, g(x)→−∞g(x)→−∞, as x→∞x→∞, g(x)→∞g(x)→∞.
45.
x-intercepts (3,0)(3,0) with multiplicity 3, (2,0)(2,0) with multiplicity 2, y-y- intercept (0,–108)(0,–108). As x→−∞x→−∞, k(x)→−∞k(x)→−∞, as x→∞x→∞, k(x)→∞.k(x)→∞.
47.
x-intercepts (0, 0),(–2, 0),(4, 0)(0, 0),(–2, 0),(4, 0) with multiplicity 1, yy-intercept (0, 0).(0, 0). As x→−∞x→−∞, n(x)→∞n(x)→∞, as x→∞x→∞, n(x)→−∞.n(x)→−∞.
49.
f(x)=−29(x−3)(x+1)(x+3)f(x)=−29(x−3)(x+1)(x+3)
51.
f(x)=14(x+2)2(x−3)f(x)=14(x+2)2(x−3)
53.
–4, –2, 1, 3 with multiplicity 1
55.
–2, 3 each with multiplicity 2
57.
f(x)=−23(x+2)(x−1)(x−3)f(x)=−23(x+2)(x−1)(x−3)
59.
f(x)=13(x−3)2(x−1)2(x+3)f(x)=13(x−3)2(x−1)2(x+3)
61.
f(x)=−15(x−1)2(x−3)3f(x)=−15(x−1)2(x−3)3
63.
f(x)=−2(x+3)(x+2)(x−1)f(x)=−2(x+3)(x+2)(x−1)
65.
f(x)=−32(2x−1)2(x−6)(x+2)f(x)=−32(2x−1)2(x−6)(x+2)
67.
local max (–.58, –.62),(–.58, –.62), local min (.58, –1.38)(.58, –1.38)
69.
global min (–.63, –.47)(–.63, –.47)
71.
global min (.75, .89)(.75, .89)
73.
f(x)=(x−500)2(x+200)f(x)=(x−500)2(x+200)
75.
f(x)=4x3−36x2+80xf(x)=4x3−36x2+80x
77.
f(x)=4x3−36x2+60x+100f(x)=4x3−36x2+60x+100
79.
f(x)=9π(x3+5x2+8x+4)f(x)=9π(x3+5x2+8x+4)
3.5 Section Exercises
1.
The binomial is a factor of the polynomial.
3.
x+6+5x−1x+6+5x-1, quotient: x+6x+6, remainder: 55
5.
3x+23x+2, quotient: 3x+23x+2, remainder: 00
7.
x−5x−5, quotient: x−5x−5, remainder: 00
9.
2x−7+16x+22x−7+16x+2, quotient: 2x−72x−7, remainder: 1616
11.
x−2+63x+1x−2+63x+1, quotient: x−2x−2, remainder: 66
13.
2x2−3x+52x2−3x+5, quotient: 2x2−3x+52x2−3x+5, remainder: 00
15.
2x2+2x+1+10x−42x2+2x+1+10x−4
17.
2x2−7x+1−22x+12x2−7x+1−22x+1
19.
3x2−11x+34−106x+33x2−11x+34−106x+3
21.
x2+5x+1x2+5x+1
23.
4x2−21x+84−323x+44x2−21x+84−323x+4
25.
x2−14x+49x2−14x+49
27.
3x2+x+23x−13x2+x+23x−1
29.
x3−3x+1x3−3x+1
31.
x3−x2+2x3−x2+2
33.
x3−6x2+12x−8x3−6x2+12x−8
35.
x3−9x2+27x−27x3−9x2+27x−27
37.
2x3−2x+22x3−2x+2
39.
Yes (x−2)(3x3−5)(x−2)(3x3−5)
41.
Yes (x−2)(4x3+8x2+x+2)(x−2)(4x3+8x2+x+2)
43.
No
45.
(x−1)(x2+2x+4)(x−1)(x2+2x+4)
47.
(x−5)(x2+x+1)(x−5)(x2+x+1)
49.
Quotient: 4x2+8x+164x2+8x+16, remainder: −1−1
51.
Quotient: 3x2+3x+53x2+3x+5, remainder: 00
53.
Quotient: x3−2x2+4x−8x3−2x2+4x−8, remainder: −6−6
55.
x6−x5+x4−x3+x2−x+1x6−x5+x4−x3+x2−x+1
57.
x3−x2+x−1+1x+1x3−x2+x−1+1x+1
59.
1+1+ix−i1+1+ix−i
61.
1+1−ix+i1+1−ix+i
63.
x2−ix−1+1−ix−ix2−ix−1+1−ix−i
65.
2x2+32x2+3
67.
2x+32x+3
69.
x+2x+2
71.
x−3x−3
73.
3x2−23x2−2
3.6 Section Exercises
1.
The theorem can be used to evaluate a polynomial.
3.
Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.
5.
Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.
7.
−106−106
9.
00
11.
255255
13.
−1−1
15.
−2,1,12−2,1,12
17.
−2−2
19.
−3−3
21.
−52,6–√,−6–√−52,6,−6
23.
2,−4,−322,−4,−32
25.
4,−4,−54,−4,−5
27.
5,−3,−125,−3,−12
29.
12,1+5√2,1−5√212,1+52,1−52
31.
3232
33.
2,3,−1,−22,3,−1,−2
35.
12,−12,2,−312,−12,2,−3
37.
−1,−1,5–√,−5–√−1,−1,5,−5
39.
−34,−12−34,−12
41.
2,3+2i,3−2i2,3+2i,3−2i
43.
−23,1+2i,1−2i−23,1+2i,1−2i
45.
−12,1+4i,1−4i−12,1+4i,1−4i
47.
1 positive, 1 negative
49.
3 or 1 positive, 0 negative
51.
0 positive, 3 or 1 negative
53.
2 or 0 positive, 2 or 0 negative
55.
2 or 0 positive, 2 or 0 negative
57.
±5,±1,±52±5,±1,±52
59.
±1,±12,±13,±16±1,±12,±13,±16
61.
1,12,−131,12,−13
63.
2,14,−322,14,−32
65.
5454
67.
f(x)=49(x3+x2−x−1)f(x)=49(x3+x2−x−1)
69.
f(x)=−15(4x3−x)f(x)=−15(4x3−x)
71.
8 by 4 by 6 inches
73.
5.5 by 4.5 by 3.5 inches
75.
8 by 5 by 3 inches
77.
Radius = 6 meters, Height = 2 meters
79.
Radius = 2.5 meters, Height = 4.5 meters
3.7 Section Exercises
1.
The rational function will be represented by a quotient of polynomial functions.
3.
The numerator and denominator must have a common factor.
5.
Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator.
7.
All reals x≠–1,1All reals x≠–1,1
9.
All reals x≠–1,–2,1,2All reals x≠–1,–2,1,2
11.
V.A. at x=–25;x=–25; H.A. at y=0;y=0; Domain is all reals x≠–25x≠–25
13.
V.A. at x=4,–9;x=4,–9; H.A. at y=0;y=0; Domain is all reals x≠4,–9x≠4,–9
15.
V.A. at x=0,4,−4;x=0,4,−4; H.A. at y=0;y=0; Domain is all reals x≠0,4,–4x≠0,4,–4
17.
V.A. at x=5;x=5; H.A. at y=0;y=0; Domain is all reals x≠5,−5x≠5,−5
19.
V.A. at x=13;x=13; H.A. at y=−23;y=−23; Domain is all reals x≠13.x≠13.
21.
none
23.
x-intercepts none, y-intercept (0,14)x-intercepts none, y-intercept (0,14)
25.
Local behavior: x→−12+,f(x)→−∞,x→−12−,f(x)→∞x→−12+,f(x)→−∞,x→−12−,f(x)→∞
End behavior: x→±∞,f(x)→12x→±∞,f(x)→12
27.
Local behavior: x→6+,f(x)→−∞,x→6−,f(x)→∞,x→6+,f(x)→−∞,x→6−,f(x)→∞, End behavior: x→±∞,f(x)→−2x→±∞,f(x)→−2
29.
Local behavior: x→13+,f(x)→∞,x→13−,x→13+,f(x)→∞,x→13−, f(x)→−∞,x→−52−,f(x)→∞,x→−52+f(x)→−∞,x→−52−,f(x)→∞,x→−52+, f(x)→−∞f(x)→−∞
End behavior: x→±∞,x→±∞, f(x)→13f(x)→13
31.
y=2x+4y=2x+4
33.
y=2xy=2x
35.
V.A.x=0,H.A.y=2V.A.x=0,H.A.y=2
37.
V.A.x=2,H.A.y=0V.A.x=2,H.A.y=0
39.
V.A.x=−4,H.A.y=2;(32,0);(0,−34)V.A.x=−4,H.A.y=2;(32,0);(0,−34)
41.
V.A.x=2,H.A.y=0,(0,1)V.A.x=2,H.A.y=0,(0,1)
43.
V.A.x=−4,x=43,H.A.y=1;(5,0);(−13,0);(0,516)V.A.x=−4,x=43,H.A.y=1;(5,0);(−13,0);(0,516)
45.
V.A.x=−1,H.A.y=1;(−3,0);(0,3)V.A.x=−1,H.A.y=1;(−3,0);(0,3); removable discontinuity (hole) at (1,2)(1,2)
47.
V.A.x=4,S.A.y=2x+9;(−1,0);(12,0);(0,14)V.A.x=4,S.A.y=2x+9;(−1,0);(12,0);(0,14)
49.
V.A.x=−2,x=4,H.A.y=1,(1,0);(5,0);(−3,0);(0,−1516)V.A.x=−2,x=4,H.A.y=1,(1,0);(5,0);(−3,0);(0,−1516)
51.
y=50x2−x−2x2−25y=50x2−x−2x2−25
53.
y=7x2+2x−24x2+9x+20y=7x2+2x−24x2+9x+20
55.
y=12x2−4x+4x+1y=12x2−4x+4x+1
57.
y=4x−3x2−x−12y=4x−3x2−x−12
59.
y=−9x−2x2−9y=−9x−2x2−9
61.
y=13x2+x−6x−1y=13x2+x−6x−1
63.
y=−6(x−1)2(x+3)(x−2)2y=−6(x−1)2(x+3)(x−2)2
65.
| xx | 2.01 | 2.001 | 2.0001 | 1.99 | 1.999 |
| yy | 100 | 1,000 | 10,000 | –100 | –1,000 |
| xx | 10 | 100 | 1,000 | 10,000 | 100,000 |
| yy | .125 | .0102 | .001 | .0001 | .00001 |
Vertical asymptote x=2,x=2, Horizontal asymptote y=0y=0
67.
| xx | –4.1 | –4.01 | –4.001 | –3.99 | –3.999 |
| yy | 82 | 802 | 8,002 | –798 | –7998 |
| xx | 10 | 100 | 1,000 | 10,000 | 100,000 |
| yy | 1.4286 | 1.9331 | 1.992 | 1.9992 | 1.999992 |
Vertical asymptote x=−4,x=−4, Horizontal asymptote y=2y=2
69.
| xx | –.9 | –.99 | –.999 | –1.1 | –1.01 |
| yy | 81 | 9,801 | 998,001 | 121 | 10,201 |
| xx | 10 | 100 | 1,000 | 10,000 | 100,000 |
| yy | .82645 | .9803 | .998 | .9998 |
Vertical asymptote x=−1,x=−1, Horizontal asymptote y=1y=1
71.
(32,∞)(32,∞)
73.
(−2,1)∪(4,∞)(−2,1)∪(4,∞)
75.
(2,4)(2,4)
77.
(2,5)(2,5)
79.
(–1,1)(–1,1)
81.
C(t)=8+2t300+20tC(t)=8+2t300+20t
83.
After about 6.12 hours.
85.
A(x)=50x2+800x.A(x)=50x2+800x. 2 by 2 by 5 feet.
87.
A(x)=πx2+100x.A(x)=πx2+100x. Radius = 2.52 meters.
3.8 Section Exercises
1.
It can be too difficult or impossible to solve for xx in terms of y.y.
3.
We will need a restriction on the domain of the answer.
5.
f−1(x)=x−−√+4f−1(x)=x+4
7.
f−1(x)=x+3−−−−√−1f−1(x)=x+3−1
9.
f−1(x)=−x−53−−−√f−1(x)=−x−53
11.
f(x)=9−x−−−−√f(x)=9−x
13.
f−1(x)=x−5−−−−√3f−1(x)=x−53
15.
f−1(x)=4−x−−−−√3f−1(x)=4−x3
17.
f−1(x)=x2−12,[0,∞)f−1(x)=x2−12,[ 0,∞ )
19.
f−1(x)=(x−9)2+44,[9,∞)f−1(x)=(x−9)2+44,[ 9,∞ )
21.
f−1(x)=(x−92)3f−1(x)=(x−92)3
23.
f−1(x)=2−8xxf−1(x)=2−8xx
25.
f−1(x)=7x−31−xf−1(x)=7x−31−x
27.
f−1(x)=5x−44x+3f−1(x)=5x−44x+3
29.
f−1(x)=x+1−−−−√−1f−1(x)=x+1−1
31.
f−1(x)=x+6−−−−√+3f−1(x)=x+6+3
33.
f−1(x)=4−x−−−−√f−1(x)=4−x
35.
f−1(x)=x−−√+4f−1(x)=x+4
37.
f−1(x)=1−x−−−−√3f−1(x)=1−x3
39.
f−1(x)=x+8−−−−√+3f−1(x)=x+8+3
41.
f−1(x)=1x−−√f−1(x)=1x
43.
[−2,1)∪[3,∞)[−2,1)∪[3,∞)
45.
[−4,2)∪[5,∞)[−4,2)∪[5,∞)
47.
(–2,0);(4,2);(22,3)(–2,0);(4,2);(22,3)
49.
(–4,0);(0,1);(10,2)(–4,0);(0,1);(10,2)
51.
(–3,−1);(1,0);(7,1)(–3,−1);(1,0);(7,1)
53.
f−1(x)=x+b24−−−−−√−b2f−1(x)=x+b24−b2
55.
f−1(x)=x3−baf−1(x)=x3−ba
57.
t(h)=200−h4.9−−−−−√,t(h)=200−h4.9, 5.53 seconds
59.
r(V)=3V4π−−−√3,r(V)=3V4π3, 3.63 feet
61.
n(C)=100C−250.6−C,n(C)=100C−250.6−C, 250 mL
63.
r(V)=V6π−−√,r(V)=V6π, 3.99 meters
65.
r(V)=V4π−−√,r(V)=V4π, 1.99 inches
3.9 Section Exercises
1.
The graph will have the appearance of a power function.
3.
No. Multiple variables may jointly vary.
5.
y=5x2y=5x2
7.
y=11944x3y=11944x3
9.
y=6x4y=6x4
11.
y=18x2y=18x2
13.
y=81x4y=81x4
15.
y=20x√3y=20x3
17.
y=10xzwy=10xzw
19.
y=10xz√y=10xz
21.
y=4xzwy=4xzw
23.
y=40xzw√t2y=40xzwt2
25.
y=256y=256
27.
y=6y=6
29.
y=6y=6
31.
y=27y=27
33.
y=3y=3
35.
y=18y=18
37.
y=90y=90
39.
y=812y=812
41.
y=34x2y=34x2
43.
y=13x−−√y=13x
45.
y=4x2y=4x2
47.
≈ 1.89 years
49.
≈ 0.61 years
51.
3 seconds
53.
48 inches
55.
≈ 49.75 pounds
57.
≈ 33.33 amperes
59.
≈ 2.88 inches
Review Exercises
1.
2−2i2−2i
3.
24+3i24+3i
5.
{2+i,2−i}{2+i,2−i}
7.
f(x)=(x−2)2−9vertex(2,–9),intercepts(5,0);(–1,0);(0,–5)f(x)=(x−2)2−9vertex(2,–9),intercepts(5,0);(–1,0);(0,–5)
9.
f(x)=325(x+2)2+3f(x)=325(x+2)2+3
11.
300 meters by 150 meters, the longer side parallel to river.
13.
Yes, degree = 5, leading coefficient = 4
15.
Yes, degree = 4, leading coefficient = 1
17.
Asx→−∞,f(x)→−∞,asx→∞,f(x)→∞Asx→−∞,f(x)→−∞,asx→∞,f(x)→∞
19.
–3 with multiplicity 2, −12−12 with multiplicity 1, –1 with multiplicity 3
21.
4 with multiplicity 1
23.
1212 with multiplicity 1, 3 with multiplicity 3
25.
x2+4x2+4with remainder 12
27.
x2−5x+20−61x+3x2−5x+20−61x+3
29.
2x2−2x−32x2−2x−3, so factored form is (x+4)(2x2−2x−3)(x+4)(2x2−2x−3)
31.
{−2,4,−12}{ −2,4,−12 }
33.
{1,3,4,12}{ 1,3,4,12 }
35.
0 or 2 positive, 1 negative
37.
Intercepts (–2,0)and(0,−25)(–2,0)and(0,−25), Asymptotes x=5x=5 and y=1.y=1.
39.
Intercepts (3, 0), (-3, 0), and (0,272)(0,272), Asymptotes x=1,x=–2,y=3.x=1,x=–2,y=3.
41.
y=x−2y=x−2
43.
f−1(x)=x−−√+2f−1(x)=x+2
45.
f−1(x)=x+11−−−−−√−3f−1(x)=x+11−3
47.
f−1(x)=(x+3)2−54,x≥−3f−1(x)=(x+3)2−54,x≥−3
49.
y=64y=64
51.
y=72y=72
53.
148.5 pounds
Practice Test
1.
20−10i20−10i
3.
{2+3i,2−3i}{2+3i,2−3i}
5.
Asx→−∞,f(x)→−∞,asx→∞,f(x)→∞Asx→−∞,f(x)→−∞,asx→∞,f(x)→∞
7.
f(x)=(x+1)2−9f(x)=(x+1)2−9, vertex (−1,−9)(−1,−9), intercepts (2,0);(−4,0);(0,−8)(2,0);(−4,0);(0,−8)
9.
60,000 square feet
11.
0 with multiplicity 4, 3 with multiplicity 2
13.
2x2−4x+11−26x+22x2−4x+11−26x+2
15.
2x2−x−42x2−x−4. So factored form is (x+3)(2x2−x−4)(x+3)(2x2−x−4)
17.
−12−12 (has multiplicity 2), −1±i15√2−1±i152
19.
−2−2 (has multiplicity 3), ±i±i
21.
f(x)=2(2x−1)3(x+3)f(x)=2(2x−1)3(x+3)
23.
Intercepts (−4,0),(0,−43)(−4,0),(0,−43), Asymptotes x=3,x=−1,y=0x=3,x=−1,y=0.
25.
y=x+4y=x+4
27.
f−1(x)=x+43−−−√3f−1(x)=x+433
29.
y=18y=18
31.
4 seconds
