Answers to the Sample for the Second Examination

1. a) $r+s=\left( -2+3i\right) +\left( 4-i\right) =\allowbreak 2+2i$

b) $r-s=\left( -2+3i\right) -\left( 4-i\right) =\allowbreak -6+4i$

c) $rs=\left( -2+3i\right) \left( 4-i\right) =\allowbreak -5+14i$

d) $\frac{r}{s}=\frac{-2+3i}{4-i}=\allowbreak -\frac{11}{17}+\frac{10}{17}i$

e) $\frac{s}{r}=\frac{4-i}{-2+3i}=\allowbreak -\frac{11}{13}-\frac{10}{13}i$

f) $\frac{r}{s}\cdot \frac{s}{r}=\left( -\frac{11}{17}+\frac{10}{17}%% i\right) \left( -\frac{11}{13}-\frac{10}{13}i\right) =\allowbreak 1$

g) $\overline{r}=\overline{\left( -2+3i\right) }=-2-3i$         

h) $r\overline{r}=\left( -2+3i\right) \left( -2-3i\right) =\allowbreak 13$

i) $\overline{s}=\overline{\left( 4-i\right) }=4+i$         

j) $s\overline{s}=\left( 4-i\right) \left( 4+i\right) =\allowbreak 17$

k) $rs-\overline{r}=$ $\left( -5+14i\right) -\left( -2-3i\right) =\allowbreak -3+17i$        

l) $r\overline{s}=\left( -2+3i\right) \left( 4+i\right) =\allowbreak -11+10i$

2.

a) $r+s=\left( 5-3i\right) +\left( -2i\right) =\allowbreak 5-5i$

b) $r-s=\left( 5-3i\right) -\left( -2i\right) =\allowbreak 5-i$

c) $rs=\left( 5-3i\right) \left( -2i\right) =\allowbreak -6-10i$

d) $\frac{r}{s}=\frac{5-3i}{-2i}=\allowbreak \frac{3}{2}+\frac{5}{2}i$

e) $\frac{s}{r}=\frac{-2i}{5-3i}=\allowbreak \frac{3}{17}-\frac{5}{17}i$

f) $\frac{r}{s}\cdot \frac{s}{r}=\left( \allowbreak \frac{3}{2}+\frac{5}{2}%% i\right) \left( \frac{3}{17}-\frac{5}{17}i\right) =\allowbreak 1$

g) $\overline{r}=\overline{\left( 5-3i\right) }=5+3i$         

h) $r\overline{r}=\left( 5-3i\right) \left( 5+3i\right) =\allowbreak 34$

i) $\overline{s}=\overline{\left( -2i\right) }=2i$        

j) $s\overline{s}=\left( -2i\right) \left( 2i\right) =\allowbreak 4$

k) $rs-\overline{r}=$ $\left( -6-10i\right) -\left( 5+3i\right) =\allowbreak -11-13i$        

l) $r\overline{s}=\left( 5-3i\right) \left( 2i\right) =\allowbreak 6+10i$

3.

a) $r+s=\left( 2\right) +\left( 3i\right) =\allowbreak 2+3i\qquad \qquad$

b) $r-s=\left( 2\right) -\left( 3i\right) =\allowbreak 2-3i$

c) $rs=\left( 2\right) \left( 3i\right) =\allowbreak 6i\qquad \qquad$

d) $\frac{r}{s}=\frac{2}{3i}=\allowbreak -\frac{2}{3}i\qquad \qquad$

e) $\frac{s}{r}=\frac{3i}{2}=\allowbreak \frac{3}{2}i$

f) $\frac{r}{s}\cdot \frac{s}{r}=\left( -\frac{2}{3}i\right) \left( \allowbreak \frac{3}{2}i\right) =\allowbreak 1\qquad \qquad$

g) $\overline{r}=\overline{\left( 2\right) }=2$                 

h) $r\overline{r}=\left( 2\right) \left( 2\right) =4$

i) $\overline{s}=\overline{\left( 3i\right) }=-3i$                 

j) $s\overline{s}=\left( 3i\right) \left( -3i\right) =\allowbreak 9\qquad \qquad$

k) $rs-\overline{r}=$ $\left( 6i\right) -\left( 2\right) =\allowbreak -2+6i$                

l) $r\overline{s}=\left( 2\right) \left( -3i\right) =\allowbreak -6i$

4. Solve for all real values of $x$:

a) $x=\frac{3}{2}\pm \frac{1}{2}\sqrt{37}$

b) $x=\frac{1}{2}\pm \frac{1}{2}\sqrt{13}$         

c) $x=3$        

d) No real solutions. $(x=-\frac{1}{2}\pm \frac{1}{2}i\sqrt{3})$

e) $x=1-\sqrt{7}$         

f) No solutions         

g) $x=4$        

h) $x=4$ is the only real solution

i) Corrected problem: $x^{6}+7x^{3}-8=0;$ The only real solutions are $x=1$ and $x=-2\allowbreak$. (There are also four complex solutions.)

j) $x=\pm 2$ are the only real solutions. ( $x=\pm i\sqrt{3}$ are also solutions)

k) $\lbrack -6,2\rbrack$

l) $(-\infty ,-8)\cup (3,\infty )$        

m) $x=-3$ is the only solution

n) $(-\infty ,-\frac{1}{2}-\frac{1}{2}\sqrt{13}\rbrack \cup \lbrack -\frac{%% 1}{2}+\frac{1}{2}\sqrt{13},\infty )$         

o) $(-\infty ,\infty )$        

p) There are no solutions

5. a) $x=1$ $$and $x=\frac{5}{2}$         

b) $x=\frac{7\pm \sqrt{89}}{4}$        

c) $x=\frac{-1\pm \sqrt{21}}{2}$        

d) $x=\frac{-1\pm i\sqrt{19}}{2}$

6. a) $-\frac{7}{3}+(-1)+(-8)+1=\allowbreak -\frac{31}{3}$        

b) $-3+(-\frac{1}{2})+(-8)+1=\allowbreak -\frac{21}{2}$

c) $(-\frac{9}{2}+\frac{1}{2}\sqrt{57})+(-\frac{9}{2}-\frac{1}{2}\sqrt{57}%% )+(-\... ...+\frac{1}{2}i\sqrt{11})+(-\frac{1}{2}-\frac{1}{2}i\sqrt{11}%% )=\allowbreak -10$

7. a) $(x-4)(x+2)=\allowbreak x^{2}-2x-8$; $$so $\ x^{2}-2x-8=0$ is the desired equation        

b) $(x-\frac{5}{2})(x+\frac{1}{6})=\allowbreak x^{2}-\frac{7}{3}x-\frac{5}{%% 12}$; so $12x^{2}-28x-5=0$ is the desired equation

c) $(x+\frac{3}{2})(x+4)=\allowbreak x^{2}+\frac{11}{2}x+6$; so $%% 2x^{2}+11x+12=0$ is the desired equation

d) $(x-3-\sqrt{5})(x-3+\sqrt{5})=\allowbreak x^{2}-6x+4$; so $%% x^{2}-6x+4=0$ is the desired equation         

e) $(x+4-3i)(x+4+3i)=\allowbreak x^{2}+8x+25$; so $x^{2}+8x+25=0$ is the desired equation         

f) $(x-\sqrt{7}i)(x+\sqrt{7}i)=\allowbreak x^{2}+7$; so $x^{2}+7=0$ is the desired equation

g) $\left( x+11\right) ^{2}=\allowbreak x^{2}+22x+121$; so $%% x^{2}+22x+121=0$ is the desired equation         

h) $(x-\frac{5}{7})^{2}=\allowbreak x^{2}-\frac{10}{7}x+\frac{25}{49}$; so $49x^{2}-70x+25=0$ is the desired equation         

8. a) $x\left( x-13\right) =30$. Therefore $x=-2$ or $%% x=15$

b) $3n\left( n+1\right) =2\left( n+2\right) ^{2}+58$. The numbers are either $-6,$ $-5$ $$and $-4$ $$or $11$, $12$ and $13$

c) $n(n+2)=4(n+n+2)-8$. The numbers are either 0 and $2$ or they are $6$ and $8$.

d) Let $r$ be the rate of the faster train in miles per hour. $\frac{240}{r}=\frac{240}{r-24}-\frac{1}{2}$.

The faster train travels at $120$ mph, and the other at $96$ mph.

e) Let $r$ be the rate of the faster train in miles per hour. $\frac{5}{2}r=4(r-30)$.

The faster train travels at $80$ mph and the stations are $200$ miles apart.

f) $\frac{1}{\frac{1}{j}+\frac{1}{j+15}}=j-5$. It would take John $15$ to complete the job is he works alone. (It would take Sally $30$ minutes working alone; and they can do the job together in $10$ minutes.)

g) $h(t)=240+32t-16t^{2}$; $v(t)=32-32t$

        i) $h\left( t\right) =240+32t-16t^{2}=0$; $$it takes $5$ seconds to hit the ground.

        ii) $v\left( t\right) =32-32t=0$; $t=1$; $h\left( 1\right) =240+32-16=256$. It is $256$ feet high at its highest point.

        iii) 0. At its highest point, the velocity is 0.

        iv) $v\left( 5\right) =32-32\cdot 5=-128$. When it hits the ground, it is moving downward with a velocity of $128$ feet per second.

        v) $h(t)=240+32t-16t^{2}=192$; $t=3$; It takes $3$ seconds to reach the window. At that thim, it is moving downward at a rate of $%% v\left( 3\right) =32-32\cdot 3=-64$ feet per second.

9. a) $x^{2}+(y-4)^{2}=4$.

This is a circle with center $\left( 0,4\right)$ and radius $2$.

b) $y=\frac{5}{9}x+\frac{10}{9}$.

This is a straight line with slope $\frac{5}{9}$ and $y$-intercept $\frac{%% 10}{9}$.

c) $\left( x-3\right) ^{2}=\frac{4}{3}(y+3)$.

This is a parabola opening upward, with vertex $(3,-3)$ and axis of symmetry $x=3$.

d) $\left( y+3\right) ^{2}=2(x+\frac{11}{4})$.

This is a parabola opening to the right with vertex $\left( -\frac{11}{4}%% ,-3\right)$ and axis of symmetry $y=-3$.

e) $\left( x-3\right) ^{2}+y^{2}=4$

This is a circle with center $\left( 3,0\right)$ and radius $2$.

f) $\left( x-3\right) ^{2}+y^{2}=0$.

This is a point circle. The unique point is $\left( 3,0\right)$.

g) $\left( x-3\right) ^{2}+y^{2}=-5$.

This is an imaginary circle.

h) $x^{3}+y^{3}+3x-5y+1=0$.

This is not a quadratic equation. Its graph is not a conic section.

i) $\left( x-2^{2}\right) +\left( y+3\right) ^{2}=13$.

This is a circle with center $\left( 2,-3\right)$ and radius $\sqrt{13}$.

10. a) $\left( x-1\right) ^{2}+\left( y-5\right) ^{2}=25$

b) $x^{2}+\left( y-1\right) ^{2}=18$

c) $x^{2}+\left( y+4\right) ^{2}=25$

d) $\left( x+3\right) ^{2}+\left( y-7\right) ^{2}=25$

e) $\left( y-1\right) ^{2}=-12(x+1)$

f) $\left( y+3\right) ^{2}=-16\left( x-1\right)$

11. a) Center $\left( 1,-2\right)$; radius $\sqrt{10}$

b) Center $\left( \frac{5}{2},\frac{1}{2}\right)$; radius $\frac{5}{2}\sqrt{2}$

c) Vertex $\left( 7,-3\right)$; focus $\left( 10,-3\right)$; directrix $x=4$; axis of symmetry $y=-3$; there is no $y$-intercept; $x$-intercept $\left( 93,0\right)$

d) Vertex $\left( 2,2\right)$; focus $\left( 2,-2\right)$; directrix $y=6$; axis of symmetry $x=2$; $y$-intercept $\left( 0,%% \frac{7}{4}\right)$; $x$-intercepts $\left( 2\pm 4\sqrt{2},0\right)$

e) Vertex $\left( -\frac{21}{8},\frac{5}{2}\right)$; focus $\left( -%% \frac{17}{8},\frac{5}{2}\right)$; directrix $x=-\frac{25}{8}$; axis of symmetry $y=\frac{5}{2}$; $y$-intercept $\left( 0,\frac{5+\sqrt{21}}{%% 2}\right)$; $x$-intercept $\left( \frac{1}{2},0\right)$

f) Slope $\frac{3}{4}$; $y$-intercept $\left( 0,-3\right)$

12. a) $\lbrack 7,\infty )$

b) $\left( -\infty ,-7\right) \cup \left( 3,\infty \right)$

c) $\left( -\infty ,\infty \right)$

d) $\lbrack -2\sqrt{2},2)\cup (2,2\sqrt{2}\rbrack$

13. a) $(h\circ g\circ f)(x)=\sqrt[3]{\frac{1}{x^{3}}}=\frac{1}{x}$

b) $(h\circ g\circ f)(x)=\sqrt[3]{\frac{1}{x^{3}-1+1}+1}=\sqrt[3]{\frac{1}{x^{3}}+1}=\frac{\sqrt[3]{1+x^{3}}}{x}$

c) $(h\circ g\circ f)(x)=\sqrt{\frac{1}{x^{2}-2+3}+2}=\sqrt{\frac{1}{%% x^{2}+1}+2}=\sqrt{\frac{2x^{2}+3}{x^{2}+1}}$

d) $(h\circ g\circ f)(x)=\frac{1}{\sqrt{x^{2}-2+2}+3}=\frac{1}{x+3}$

e) $(h\circ g\circ f)(x)=\frac{1}{\left( \sqrt{x+2}\right) ^{2}-2+3}=\frac{%% 1}{x+3}$

f) $(h\circ g\circ f)(x)=\left( \sqrt{\frac{1}{x+3}+2}\right) ^{2}-2=%% \frac{1}{x+3}$.

&lsqb#lbrack;Note, the second $g\left( x\right)$ in part f) of the problem should have been $h\left( x\right)$.&rsqb#rbrack;

14. $f(x)=(x+1)^{3}$, $g(x)=\frac{1}{\sqrt[3]{x}}$, and $%% h(x)=2x-1$

a) $\frac{1}{x+1}=\left( g\circ f\right) \left( x\right)$

b) $8x^{3}=\left( f\circ h\right) \left( x\right)$

c) $\frac{1}{\sqrt[3]{2x^{3}+6x^{2}+6x+1}}=\left( g\circ h\circ f\right) \left( x\right)$

d) $\frac{1}{\sqrt[3]{4x^{2}-4x+1}}=\left( g\circ (h\cdot h)\right) \left( x\right)$ &lsqb#lbrack;This was a poorly conceived problem. Since $\cdot$ refers to multiplication of functions and not to composition, this doesn't really follow the instructions for the problem.&rsqb#rbrack;

e) $4x-3=\left( h\circ h\right) \left( x\right)$

f) $\sqrt[9]{2x-1}=\left( g\circ g\circ h\right) \left( x\right)$

15. For each of the following functions, determine whether or not it has an inverse. If it does have an inverse, find the inverse function as well as its domain and range. If it doesn't have an inverse you must explain why not.

a) $f^{-1}\left( x\right) =\frac{2x-1}{2}$

b) $g^{-1}(x)=\sqrt[3]{x^{3}+8}$

c) $g^{-1}(x)=\sqrt[3]{x^{2}+8}$, for $x\geq 0$

d) $h(x)=x^{2}-5x+4$ does not have an inverse. It is not one-to-one. For example, $h\left( 1\right) =0=h\left( 4\right)$                          

e) $k^{-1}(x)=\frac{3x+5}{3-x}$

f) $f^{-1}(x)=\sqrt[3]{x-1}\vspace{1pt}$        

g) $f^{-1}(x)=\frac{x-1}{x}$

h) $k^{-1}(x)=\frac{1}{4-x}$

16.

a) b)

17. and 18.

a) $p(x)=\allowbreak x^{3}+x^{2}-8x-12$ is not symmetric about either the $y$-axis or the origin.

Its $y$-intercept is $\left( 0,-12\right)$, and its $x$-intercepts are $%% \left( -2,0\right)$ and $\left( 3,0\right)$.

It has no asymptotes.

The function does not have an inverse.

b) $p(x)=\allowbreak x^{3}+3x^{2}-9x+5$ is not symmetric about either the $y$-axis or the origin.

Its $y$-intercept is $\left( 0,5\right)$, and its $x$-intercepts are $%% \left( -5,0\right)$ and $\left( 1,0\right)$.

It has no asymptotes.

The function does not have an inverse.

c) $p(x)=\allowbreak x^{3}+6x^{2}+3x-10$ is not symmetric about either the $y$-axis or the origin.

Its $y$-intercept is $\left( 0,-10\right)$, and its $x$-intercepts are $%% \left( -5,0\right)$, $\left( -2,0\right)$ and $\left( 1,0\right)$.

It has no asymptotes.

The function does not have an inverse.

d) $q(x)=\frac{x^{2}-7x+6}{x^{2}-4}$ is not symmetric about either the $y$-axis or the origin.

Its $y$-intercept is $\left( 0,-\frac{3}{2}\right)$, and its $x$-intercepts are $\left( 1,0\right)$ and $\left( 6,0\right)$.

Its vertical asymptotes are $x=\pm 2$; and its horizontal asymptote is $%% y=1$.

The function does not have an inverse.

e) $q(x)=\frac{x^{2}-7x+10}{x^{2}-9}$ is not symmetric about either the $y$-axis or the origin.

Its $y$-intercept is $\left( 0,-\frac{10}{9}\right)$, and its $x$-intercepts are $\left( 2,0\right)$ and $\left( 5,0\right)$.

Its vertical asymptotes are $x=\pm 3$; and its horizontal asymptote is $%% y=1$.

The function does not have an inverse.

f) $t\left( x\right) =\sqrt{x+3}$ is not symmetric about either the $y$-axis or the origin.

Its $y$-intercept is $\left( 0,\sqrt{3}\right)$, and its $x$-intercept is $\left( -3,0\right)$.

It has no asymptotes.

The function does have an inverse. $t^{-1}\left( x\right) =x^{2}-3$, for $x\geq 0$.

g) $f(x)=\sqrt{\frac{2x^{2}-8}{x^{2}-9}}$ is symmetric about the $y$-axis. It is not symmetric about the origin.

Its $y$-intercept is $\left( 0,\sqrt{\frac{8}{9}}\right)$, and its $x$-intercepts are $\left( -2,0\right)$ and $\left( 2,0\right)$.

Its vertical asymptotes are $x=\pm 3$; and its horizontal asymptote is $%% y=2$.