14.6: Section 2.3 Answers
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1. (b) x_{0}\neq k\pi \;\;\;\;(k=\text{integer})
2. (b) (x_{0},y_{0})\neq (0,0)
3. (b) x_{0}y_{0}\neq (2k+1)\frac{\pi }{2} \;\;\;\; (k=\text{integer})
4. (b) x_{0}y_{0}>0\text{ and }x_{0}y_{0}\neq 1
5.
6. (b) all (x_{0}, y_{0})
7. (b) all (x_{0}, y_{0})
8. (b) (x_{0}, y_{0}) such that x_{0}\neq 4y_{0}
9.
10.
11. (b) all (x_{0}, y_{0})
12. (b) all (x_{0}, y_{0}) such that (x_{0}, y_{0}) >0
13. (b) all (x_{0}, y_{0}) with x_{0}\neq 1,\quad y_{0}\neq (2k+1)\frac{\pi }{2}(k=\text{integer})
16. y=\left(\frac{3}{5}x+1 \right)^{5/3},\quad -\infty <x<\infty, is a solution.
Also, y=\left\{\begin{array}{cc}{0,}&{-\infty <x\leq -\frac{5}{3}}\\[4pt]{(\frac{3}{5}x+1)^{5/3},}&{-\frac{5}{3}<x<\infty }\end{array} \right.\nonumber is a solution. For every a\geq \frac{5}{3}, the following function is also a solution: y=\left\{\begin{array}{cc}{(\frac{3}{5}(x+a))^{5/3},}&{-\infty <x<-a,}\\[4pt]{0,}&{-a\leq x\leq -\frac{5}{3}}\\[4pt]{(\frac{3}{5}x+1)^{5/3},}&{-\frac{5}{3}<x<\infty }\end{array} \right.\nonumber
17.
18. y_{1}=1; y_{2}=1+|x|^{3};y_{3}=1-|x|^{3};y_{4}=1+x^{3};y_{5}=1-x^{3}
y_{6}=\left\{\begin{array}{cc}{1+x^{3},}&{x\geq 0,}\\[4pt]{1,}&{x<0}\end{array} \right.;\quad y_{7}=\left\{\begin{array}{cc}{1-x^{3},}&{x\geq 0,}\\[4pt]{1,}&{x<0}\end{array} \right.;\nonumber
y_{8}=\left\{\begin{array}{cc}{1,}&{x\geq 0,}\\[4pt]{1+x^{3},}&{x<0}\end{array} \right.;\quad y_{9}=\left\{\begin{array}{cc}{1,}&{x\geq 0,}\\[4pt]{1-x^{3},}&{x<0}\end{array} \right.\nonumber
19. y=1+(x^{2}+4)^{3/2},\quad -\infty <x<\infty
20.