Solutions
Solutions to Try Its
1. End behavior: as x→±∞,f(x)→0; Local behavior: as 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)→∞, and as x→±∞,f(x)→−4.
The function is f(x)=(x−3)21−4.
3. 1112
4. The domain is all real numbers except x=1 and x=5.
5. Removable discontinuity at x=5. Vertical asymptotes: x=0, x=1.
6. Vertical asymptotes at x=2 and x=−3; horizontal asymptote at y=4.
7. For the transformed reciprocal squared function, we find the rational form. f(x)=(x−3)21−4=(x−3)21−4(x−3)2=(x−3)(x−3)1−4(x2−6x+9)=x2−6x+9−4x2+24x−35
Because the numerator is the same degree as the denominator we know that as x→±∞,f(x)→−4;so y=−4 is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is x=3, because as x→3,f(x)→∞. We then set the numerator equal to 0 and find the x-intercepts are at (2.5,0) and (3.5,0). Finally, we evaluate the function at 0 and find the y-intercept to be at (0,9−35).
8. Horizontal asymptote at y=21. Vertical asymptotes at x=1andx=3. y-intercept at (0,34.)
x-intercepts at (2,0) and (−2,0). (−2,0) is a zero with multiplicity 2, and the graph bounces off the x-axis at this point. (2,0) is a single zero and the graph crosses the axis at this point.
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 x=−1,1
9. All reals x=−1,−2,1,2
11. V.A. at x=−52; H.A. at y=0; Domain is all reals x=−52
13. V.A. at x=4,−9; H.A. at y=0; Domain is all reals x=4,−9
15. V.A. at x=0,4,−4; H.A. at y=0; Domain is all reals x=0,4,−4
17. V.A. at x=−5; H.A. at y=0; Domain is all reals x=5,−5
19. V.A. at x=31; H.A. at y=−32; Domain is all reals x=31.
21. none
23. x-intercepts none, y-intercept (0,41)
25. Local behavior: x→−21+,f(x)→−∞,x→−21−,f(x)→∞
End behavior: x→±∞,f(x)→21
27. Local behavior: x→6+,f(x)→−∞,x→6−,f(x)→∞, End behavior: x→±∞,f(x)→−2
29. Local behavior: x→−31+,f(x)→∞,x→−31−, f(x)→−∞,x→25−,f(x)→∞,x→25+ , f(x)→−∞
End behavior: x→±∞, f(x)→31
31. y=2x+4
33. y=2x
35. V.A. x=0,H.A. y=2
37. V.A. x=2, H.A. y=0
39. V.A. x=−4, H.A. y=2;(23,0);(0,−43)
41. V.A. x=2, H.A. y=0, (0,1)
43. V.A. x=−4, x=34, H.A. y=1;(5,0);(−31,0);(0,165)
45. V.A. x=−1, H.A. y=1;(−3,0);(0,3)
47. V.A. x=4, S.A. y=2x+9;(−1,0);(21,0);(0,41)
49. V.A. x=−2, x=4, H.A. y=1,(1,0);(5,0);(−3,0);(0,−1615)
51. y=50x2−25x2−x−2
53. y=7x2+9x+20x2+2x−24
55. y=21x+1x2−4x+4
57. y=4x2−x−12x−3
59. y=−9x2−9x−2
61. y=31x−1x2+x−6
63. y=−6(x+3)(x−2)2(x−1)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=2, Horizontal asymptote y=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=−4, Horizontal asymptote y=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=−1, Horizontal asymptote y=1
71. (23,∞)
73. (−2,1)∪(4,∞)
75. (2,4)
77. (2,5)
79. (−1,1)
81. C(t)=300+20t8+2t
83. After about 6.12 hours.
85. A(x)=50x2+x800. 2 by 2 by 5 feet.
87. A(x)=πx2+x100. Radius = 2.52 meters.Licenses & Attributions
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- Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175..