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13.2.3: Chapter 3

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    117278
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    Try It

    3.1 Complex Numbers

    1.

    −24−−−−√=0+2i6–√−24=0+2i6

    2.

    Graph of the plotted point, -4-i.

    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.

    1. ⓐ 3 seconds
    2. ⓑ 256 feet
    3. ⓒ 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.

    Graph of f(x)=(1/4)x(x-1)^4(x+3)^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.Graph of f(x)=1/(x-3)^2-4 with its vertical asymptote at x=3 and its horizontal asymptote at y=-4.

    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.

    Graph of f(x)=(x+2)^2(x-2)/2(x-1)^2(x-3) with its vertical and horizontal asymptotes.

    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

    3.9 Modeling Using Variation

    1.

    12831283

    2.

    9292

    3.

    x=20x=20

    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.

    Graph of the plotted point, 1-2i.

    15.

    Graph of the plotted point, i.

    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.Graph of f(x) = x^2-2x

    Vertex (1,−1),(1,−1), Axis of symmetry is x=1.x=1. Intercepts are (0,0),(2,0).(0,0),(2,0).

    55.Graph of f(x)x^2-5x-6

    Vertex (52,−494),(52,−494), Axis of symmetry is x=52,x=52, intercepts: (6,0),(−1,0).(6,0), (−1,0).

    57.Graph of f(x)=-2x^2+5x-8

    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.Graph of f(x)=x^3(x-2).

    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.Graph of f(x)=x(14-2x)(10-2x).

    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.ea64d1049ba1b4692bb5442f811950250c6fb534

    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.Graph of f(x)=x^3-27.

    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.Graph of f(x)=-x^3+x^2+2x.

    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)→∞.

    Graph of g(x)=(x+4)(x-1)^2.45.

    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)→∞.

    Graph of k(x)=(x-3)^3(x-2)^2.47.

    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)→−∞.

    Graph of n(x)=-3x(x+2)(x-4).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

    Graph of f(x)=x^4-x^2-1.49.

    3 or 1 positive, 0 negative

    Graph of f(x)=x^3-2x^2+x-1.51.

    0 positive, 3 or 1 negative

    Graph of f(x)=2x^3+37x^2+200x+300.53.

    2 or 0 positive, 2 or 0 negative

    Graph of f(x)=2x^4-5x^3-5x^2+5x+3.55.

    2 or 0 positive, 2 or 0 negative

    Graph of f(x)=10x^4-21x^2+11.57.

    ±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

    Graph of a rational function.37.

    V.A.x=2,H.A.y=0V.A.x=2,H.A.y=0

    Graph of a rational function.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)

    Graph of p(x)=(2x-3)/(x+4) with its vertical asymptote at x=-4 and horizontal asymptote at y=2.41.

    V.A.x=2,H.A.y=0,(0,1)V.A.x=2,H.A.y=0,(0,1)

    Graph of s(x)=4/(x-2)^2 with its vertical asymptote at x=2 and horizontal asymptote at y=0.43.

    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)

    Graph of f(x)=(3x^2-14x-5)/(3x^2+8x-16) with its vertical asymptotes at x=-4 and x=4/3 and horizontal asymptote at y=1.45.

    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)

    Graph of a(x)=(x^2+2x-3)/(x^2-1) with its vertical asymptote at x=-1 and horizontal asymptote at y=1.47.

    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)

    Graph of h(x)=(2x^2+x-1)/(x-1) with its vertical asymptote at x=4 and slant asymptote at y=2x+9.49.

    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)

    Graph of w(x)=(x-1)(x+3)(x-5)/(x+2)^2(x-4) with its vertical asymptotes at x=-2 and x=4 and horizontal asymptote at y=1.51.

    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,∞)

    Graph of f(x)=4/(2x-3).73.

    (−2,1)∪(4,∞)(−2,1)∪(4,∞)

    Graph of f(x)=(x+2)/(x-1)(x-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

    Graph of f(x)=4- x^2 and its inverse, f^(-1)(x)= sqrt(4-x).35.

    f−1(x)=x−−√+4f−1(x)=x+4

    Graph of f(x)= (x-4)^2 and its inverse, f^(-1)(x)= sqrt(x)+4.37.

    f−1(x)=1−x−−−−√3f−1(x)=1−x3

    Graph of f(x)= 1-x^3 and its inverse, f^(-1)(x)= (1-x)^(1/3).39.

    f−1(x)=x+8−−−−√+3f−1(x)=x+8+3

    Graph of f(x)= x^2-6x+1 and its inverse, f^(-1)(x)= sqrt(x+8)+3.41.

    f−1(x)=1x−−√f−1(x)=1x

    Graph of f(x)= 1/x^2 and its inverse, f^(-1)(x)= sqrt(1/x).43.

    [−2,1)∪[3,∞)[−2,1)∪[3,∞)

    Graph of f(x)= sqrt((x+2)(x-3)/(x-1)).45.

    [−4,2)∪[5,∞)[−4,2)∪[5,∞)

    Graph of f(x)= sqrt((x^2-x-20)/(x-2)).47.

    (–2,0);(4,2);(22,3)(–2,0);(4,2);(22,3)

    Graph of f(x)= x^3-x-2.49.

    (–4,0);(0,1);(10,2)(–4,0);(0,1);(10,2)

    Graph of f(x)= x^3+3x-4.51.

    (–3,−1);(1,0);(7,1)(–3,−1);(1,0);(7,1)

    Graph of f(x)= x^4+5x+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

    Graph of y=3/4(x^2).43.

    y=13x−−√y=13x

    Graph of y=1/3sqrt(x).45.

    y=4x2y=4x2

    Graph of y=4/(x^2).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)

    Graph of f(x)=x^2-4x-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.

    Graph of f(x)=(x+1)/(x-5).39.

    Intercepts (3, 0), (-3, 0), and (0,272)(0,272), Asymptotes x=1,x=–2,y=3.x=1,x=–2,y=3.

    Graph of f(x)=(3x^2-27)/(x^2+x-2).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)

    Graph of f(x)=x^2+2x-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.

    Graph of f(x)=(x+4)/(x^2-2x-3).25.

    y=x+4y=x+4

    27.

    f−1(x)=x+43−−−√3f−1(x)=x+433

    29.

    y=18y=18

    31.

    4 seconds


    13.2.3: Chapter 3 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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