Answer Key 11.1

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1. 1. No
   2. Yes
   3. No
   4. Yes
   5. Yes
   6. No
   7. Yes
   8. \(y^2=1+x^2\)
      \(y=\pm \sqrt{1+x^2}\)
      No
   9. \(\sqrt{y}=2-x\)
      \(y=(2-x)^2\)
      Yes
   10. \(y^2=1-x^2\)
       \(y=\pm \sqrt{1-x^2}\)
       No
2. All real numbers \(-\infty, \infty\)
3. \(\begin{array}{rrrrr}
   \\ \\ \\ \\ \\
   5&-&4x&\ge &0 \\
   -5&&&&-5 \\
   \midrule
   &&\dfrac{-4x}{-4}&\ge &\dfrac{-5}{-4} \\ \\
   &&x&\le &\dfrac{5}{4} \\
   \end{array}\)

   \(\left(-\infty, \dfrac{5}{4}\right]\)

4. \(t^2\neq 0\)
   \(t\neq \sqrt{0}\text{ or }0\)
5. All real or \((-\infty, \infty)\)
6. \(\begin{array}{rrrrr}
   \\ \\ \\ \\ \\
   t^2&+&1&\neq &0 \\
   &-&1&&-1 \\
   \midrule
   &&t^2&\neq &-1 \\
   &&t&\neq & i \\ \\
   &&t&=&\mathbb{R}
   \end{array}\)
7. \(\begin{array}{rrrrr}
   \\ \\
   x&-&16&\ge &0 \\
   &+&16&&+16 \\
   \midrule
   &&x&\ge &16 \\
   \end{array}\)

   \([16, \infty)\)

8. \(x^2-3x-4\neq 0\)
   \((x-4)(x+1)\neq 0\)
   \(x\neq 4,1\)
9. \(\begin{array}{ll}
   \\ \\ \\
   \begin{array}{rrrrr}
   \\ \\
   3x&-&12&\ge &0 \\
   &+&12&&+12 \\
   \midrule
   &&\dfrac{3x}{3}&\ge &\dfrac{12}{3} \\ \\
   &&x&\ge &4
   \end{array}
   &\hspace{0.25in}
   \begin{array}{rrl}
   \\
   x^2-25&\neq &0 \\
   (x-5)(x+5)&\neq &0 \\
   x&\neq &5, -5 \\ \\
   \therefore x&\ge &4, \neq \pm 5
   \end{array}
   \end{array}\)
10. \(\begin{array}{rrl}
    \\
    g(0)&=&\cancel{4(0)}-4 \\
    &=&-4
    \end{array}\)
11. \(\begin{array}{rrl}
    \\
    g(2)&=&-3\cdot 5^{-2} \\
    &=&-\dfrac{3}{25}
    \end{array}\)
12. \(\begin{array}{rrl}
    \\
    f(-9)&=&(-9)^2+4 \\
    &=&81+4 \\
    &=&85
    \end{array}\)
13. \(\begin{array}{rrl}
    \\
    f(10)&=&10-3 \\
    &=&7
    \end{array}\)
14. \(\begin{array}{rrl}
    \\ \\ \\ \\ \\ \\ \\ \\
    f(-2)&=&3^{-2}-2 \\ \\
    &=&\dfrac{1}{9}-2 \\ \\
    &=&\dfrac{1}{9}-\dfrac{18}{9} \\ \\
    &=&-\dfrac{17}{9}
    \end{array}\)
15. \(\begin{array}{rrl}
    \\ \\
    f(2)&=&-3^{2-1}-3 \\
    &=&-3^1-3 \\
    &=&-6
    \end{array}\)
16. \(\begin{array}{rrl}
    \\ \\ \\
    k(2)&=&-2\cdot 4^{2(2)-2} \\
    &=&-2\cdot 4^{4-2} \\
    &=&-2\cdot 4^2 \\
    &=&-32
    \end{array}\)
17. \(\begin{array}{rrl}
    \\ \\ \\ \\ \\ \\
    p(-2)&=&-2\cdot 4^{2(-2)+1}+1 \\
    &=&-2\cdot 4^{-4+1}+1 \\
    &=&-2\cdot 4^{-3}+1 \\
    &=&-\dfrac{2}{64}+1 \\ \\
    &=&-\dfrac{1}{32}+1 \Rightarrow \dfrac{-31}{32}
    \end{array}\)
18. \(\begin{array}{rrl}
    \\
    h(-4x)&=&(-4x)^3+2 \\
    &=&-64x^3+2
    \end{array}\)
19. \(\begin{array}{rrl}
    \\ \\
    h(n+2)&=&4(n+2)+2 \\
    &=&4n+8+2 \\
    &=&4n+10
    \end{array}\)
20. \(\begin{array}{rrl}
    \\ \\
    h(-1+x)&=&3(-1+x)+2 \\
    &=&-3+3x+2 \\
    &=&3x-1
    \end{array}\)
21. \(\begin{array}{rrl}
    \\ \\ \\
    h\left(\dfrac{1}{3}\right)&=&-3\cdot 2^{\frac{1}{3}+3} \\
    &=& -2^3\cdot 3\sqrt[3]{2}\\
    &=&-8\cdot 3\sqrt[3]{2} \\
    &=&-24 \sqrt[3]{2}
    \end{array}\)
22. \(\begin{array}{rrl}
    \\
    h(x^4)&=&(x^4)^2+1 \\
    &=&x^8+1
    \end{array}\)
23. \(\begin{array}{rrl}
    \\
    h(t^2)&=&(t^2)^2+t \\
    &=&t^4+t
    \end{array}\)
24. \(\begin{array}{rrl}
    \\
    f(0)&=&|\cancel{3(0)}+1|+1 \\
    &=&1+1\text{ or }2
    \end{array}\)
25. \(\begin{array}{rrl}
    \\ \\ \\
    f(-6)&=&-2 |-(-6)-2 | +1 \\
    &=&-2 |6-2| + 1 \\
    &=& -2(4)+1 \\
    &=& -8 + 1\text{ or }-7
    \end{array}\)
26. \(\begin{array}{rrl}
    \\
    f(10)&=&|10+3| \\
    &=&13
    \end{array}\)
27. \(\begin{array}{rrl}
    \\ \\
    p(5)&=&-|5|+1 \\
    &=&-5+1 \\
    &=& -4
    \end{array}\)