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Study Guides > College Algebra

Solutions

Solutions to Try Its

1. End behavior: as x±,f(x)0x\to \pm \infty , f\left(x\right)\to 0; Local behavior: as x0,f(x)x\to 0, f\left(x\right)\to \infty (there are no x- or y-intercepts) 2. The function and the asymptotes are shifted 3 units right and 4 units down. As x3,f(x)x\to 3,f\left(x\right)\to \infty\\ , and as x±,f(x)4x\to \pm \infty ,f\left(x\right)\to -4.

The function is f(x)=1(x3)24f\left(x\right)=\frac{1}{{\left(x - 3\right)}^{2}}-4.

Graph of f(x)=1/(x-3)^2-4 with its vertical asymptote at x=3 and its horizontal asymptote at y=-4. 3. 1211\frac{12}{11} 4. The domain is all real numbers except x=1x=1 and x=5x=5. 5. Removable discontinuity at x=5x=5. Vertical asymptotes: x=0, x=1x=0,\text{ }x=1. 6. Vertical asymptotes at x=2x=2 and x=3x=-3; horizontal asymptote at y=4y=4. 7. For the transformed reciprocal squared function, we find the rational form. f(x)=1(x3)24=14(x3)2(x3)2=14(x26x+9)(x3)(x3)=4x2+24x35x26x+9f\left(x\right)=\frac{1}{{\left(x - 3\right)}^{2}}-4=\frac{1 - 4{\left(x - 3\right)}^{2}}{{\left(x - 3\right)}^{2}}=\frac{1 - 4\left({x}^{2}-6x+9\right)}{\left(x - 3\right)\left(x - 3\right)}=\frac{-4{x}^{2}+24x - 35}{{x}^{2}-6x+9}

Because the numerator is the same degree as the denominator we know that as x±,f(x)4;so y=4x\to \pm \infty , f\left(x\right)\to -4; \text{so } y=-4 is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is x=3x=3, because as x3,f(x)x\to 3,f\left(x\right)\to \infty . We then set the numerator equal to 0 and find the x-intercepts are at (2.5,0)\left(2.5,0\right) and (3.5,0)\left(3.5,0\right). Finally, we evaluate the function at 0 and find the y-intercept to be at (0,359)\left(0,\frac{-35}{9}\right).

8. Horizontal asymptote at y=12y=\frac{1}{2}. Vertical asymptotes at x=1andx=3x=1 \text{and} x=3. y-intercept at (0,43.)\left(0,\frac{4}{3}.\right)

x-intercepts at (2,0) and (2,0)\left(2,0\right) \text{ and }\left(-2,0\right). (2,0)\left(-2,0\right) is a zero with multiplicity 2, and the graph bounces off the x-axis at this point. (2,0)\left(2,0\right) 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.

Solutions to Try Its

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 x1,1\text{All reals }x\ne -1, 1 9. All reals x1,2,1,2\text{All reals }x\ne -1, -2, 1, 2 11. V.A. at x=25x=-\frac{2}{5}; H.A. at y=0y=0; Domain is all reals x25x\ne -\frac{2}{5} 13. V.A. at x=4,9x=4, -9; H.A. at y=0y=0; Domain is all reals x4,9x\ne 4, -9 15. V.A. at x=0,4,4x=0, 4, -4; H.A. at y=0y=0; Domain is all reals x0,4,4x\ne 0,4, -4 17. V.A. at x=5x=-5; H.A. at y=0y=0; Domain is all reals x5,5x\ne 5,-5 19. V.A. at x=13x=\frac{1}{3}; H.A. at y=23y=-\frac{2}{3}; Domain is all reals x13x\ne \frac{1}{3}. 21. none 23. x-intercepts none, y-intercept (0,14)x\text{-intercepts none, }y\text{-intercept }\left(0,\frac{1}{4}\right) 25. Local behavior: x12+,f(x),x12,f(x)x\to -{\frac{1}{2}}^{+},f\left(x\right)\to -\infty ,x\to -{\frac{1}{2}}^{-},f\left(x\right)\to \infty End behavior: x±,f(x)12x\to \pm \infty ,f\left(x\right)\to \frac{1}{2} 27. Local behavior: x6+,f(x),x6,f(x)x\to {6}^{+},f\left(x\right)\to -\infty ,x\to {6}^{-},f\left(x\right)\to \infty , End behavior: x±,f(x)2x\to \pm \infty ,f\left(x\right)\to -2 29. Local behavior: x13+,f(x),x13x\to -{\frac{1}{3}}^{+},f\left(x\right)\to \infty ,x\to -{\frac{1}{3}}^{-}, f(x),x52,f(x),x52+f\left(x\right)\to -\infty ,x\to {\frac{5}{2}}^{-},f\left(x\right)\to \infty ,x\to {\frac{5}{2}}^{+}f(x)f\left(x\right)\to -\infty End behavior: x±x\to \pm \infty\\ , f(x)13f\left(x\right)\to \frac{1}{3} 31. y=2x+4y=2x+4 33. y=2xy=2x 35. V.A. x=0,H.A. y=2V.A.\text{ }x=0,H.A.\text{ }y=2 Graph of a rational function. 37. V.A. x=2, H.A. y=0V.A.\text{ }x=2,\text{ }H.A.\text{ }y=0 Graph of a rational function. 39. V.A. x=4, H.A. y=2;(32,0);(0,34)V.A.\text{ }x=-4,\text{ }H.A.\text{ }y=2;\left(\frac{3}{2},0\right);\left(0,-\frac{3}{4}\right) 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.\text{ }x=2,\text{ }H.A.\text{ }y=0,\text{ }\left(0,1\right) 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.\text{ }x=-4,\text{ }x=\frac{4}{3},\text{ }H.A.\text{ }y=1;\left(5,0\right);\left(-\frac{1}{3},0\right);\left(0,\frac{5}{16}\right) 45. V.A. x=1, H.A. y=1;(3,0);(0,3)V.A.\text{ }x=-1,\text{ }H.A.\text{ }y=1;\left(-3,0\right);\left(0,3\right) 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. 47. V.A. x=4, S.A. y=2x+9;(1,0);(12,0);(0,14)V.A.\text{ }x=4,\text{ }S.A.\text{ }y=2x+9;\left(-1,0\right);\left(\frac{1}{2},0\right);\left(0,\frac{1}{4}\right) 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.\text{ }x=-2,\text{ }x=4,\text{ }H.A.\text{ }y=1,\left(1,0\right);\left(5,0\right);\left(-3,0\right);\left(0,-\frac{15}{16}\right) 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=50x2x2x225y=50\frac{{x}^{2}-x - 2}{{x}^{2}-25} 53. y=7x2+2x24x2+9x+20y=7\frac{{x}^{2}+2x - 24}{{x}^{2}+9x+20} 55. y=12x24x+4x+1y=\frac{1}{2}\frac{{x}^{2}-4x+4}{x+1} 57. y=4x3x2x12y=4\frac{x - 3}{{x}^{2}-x - 12} 59. y=9x2x29y=-9\frac{x - 2}{{x}^{2}-9} 61. y=13x2+x6x1y=\frac{1}{3}\frac{{x}^{2}+x - 6}{x - 1} 63. y=6(x1)2(x+3)(x2)2y=-6\frac{{\left(x - 1\right)}^{2}}{\left(x+3\right){\left(x - 2\right)}^{2}} 65.
x 2.01 2.001 2.0001 1.99 1.999
y 100 1,000 10,000 –100 –1,000
x 10 100 1,000 10,000 100,000
y .125 .0102 .001 .0001 .00001
Vertical asymptote x=2x=2, Horizontal asymptote y=0y=0 67.
x –4.1 –4.01 –4.001 –3.99 –3.999
y 82 802 8,002 –798 –7998
x 10 100 1,000 10,000 100,000
y 1.4286 1.9331 1.992 1.9992 1.999992

Vertical asymptote x=4x=-4, Horizontal asymptote y=2y=2

69.
x –.9 –.99 –.999 –1.1 –1.01
y 81 9,801 998,001 121 10,201
x 10 100 1,000 10,000 100,000
y .82645 .9803 .998 .9998
Vertical asymptote x=1x=-1, Horizontal asymptote y=1y=1 71. (32,)\left(\frac{3}{2},\infty \right) Graph of f(x)=4/(2x-3). 73. (2,1)(4,)\left(-2,1\right)\cup \left(4,\infty \right) Graph of f(x)=(x+2)/(x-1)(x-4). 75. (2,4)\left(2,4\right) 77. (2,5)\left(2,5\right) 79. (1,1)\left(-1,\text{1}\right) 81. C(t)=8+2t300+20tC\left(t\right)=\frac{8+2t}{300+20t} 83. After about 6.12 hours. 85. A(x)=50x2+800xA\left(x\right)=50{x}^{2}+\frac{800}{x}. 2 by 2 by 5 feet. 87. A(x)=πx2+100xA\left(x\right)=\pi {x}^{2}+\frac{100}{x}. Radius = 2.52 meters.

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