\begin{homework} \
\label{homework:2}

\medskip
 

\exercise Find the domain of each of the following functions:

\begin{xalignat*}{2}
 1.\;& y=f(x)=\sqrt{2x-3}.     
&  
 7.\;& y=f(x)=\sqrt{r^2-(x-h)^2}\notag\\
&& \text{ ($r$, $h$ are positive constants)} \notag\\
 2.\;& y=f(x)=\frac{1}{ x+1}.    
&	 
 8.\;& y=f(x)=\sqrt{1-(1/x)}.\notag\\                                        
 3.\;& y=f(x)=\frac{1}{ x^2-1}.  
&	 
 9.\;& y=f(x)=\frac{1}{\sqrt{1-(3x)^2}}.\notag\\                               
 4.\;& y=f(x)=\sqrt{-1/x}.     
&	 
10.\;& y=f(x)=\sqrt{x}+\frac{1}{x-1}.\notag\\                                
 5.\;& y=f(x)={\root 3 \of x}. 
&	 
11.\;& y=f(x)=\frac{1}{\sqrt{x}-1}.\notag\\                                   
 6.\;& y=f(x)={\root 4 \of x}.  
& & \notag \\
\intertext{Starting with the graph of $y=\sqrt{x}$, the graph of $y=1/x$, and the graph of $y=\sqrt{1-x^2}$ (the upper unit semicircle), sketch the graph of each of the following functions:}
%
%\begin{xalignat*}{2}
12.\;& y=f(x)=\sqrt{x-2}.       
& 
18.\;& y=f(x)=-1-\frac{1}{x+2}. \notag \\      
13.\;&  y=f(x)=4+\sqrt{x+2}.	
& 
19.\;& y=f(x)=\frac{x}{1-x}.\notag \\          
14.\;& y=f(x)=-\sqrt{-x}.	
& 
20.\;& y=f(x)=2+\sqrt{1-(x-1)^2}.\notag \\    
15.\;& y=f(x)=-4+\sqrt{-(x-2)}.	
& 
\quad 21.\;& y=f(x)=2\sqrt{1-(x/3)^2}.\notag \\     
16.\;& y=f(x)=\frac{1}{x+1}.	
& 
22.\;& y=f(x)=4+2\sqrt{1-(x-5)^2/9}.\notag \\ 
17.\;& y=f(x)=1+\frac{1}{x-1}.	
& 
23.\;& y=f(x)=\sqrt{100-25(x-1)^2}+2.\notag \\
\intertext{In questions 24--30, $y=f(x)$ is given by the following graph:
%---------- figBE.ps---------------------------------
\begin{center}
\begin{psfigure}{2in}{1.25in}{figBE.ps}
\puttext{87,-30}{\Small$\pi$}
\puttext{60,30}{$y = f(x)$}
\end{psfigure}\\
\end{center}
%----------------------------------------
%
\noindent
Using that graph, sketch:
}
24.\;&y=f(x-1).   & 28.\;&y=2f(3(x-2))+1.     \notag\\
25.\;&y=1+f(x+2). & 29.\;&y=0.5f(3x-3).       \notag\\
26.\;&y=1+2f(x).  & 30.\;&y=f(\frac{x}{3}+1)+2.\notag\\
27.\;&y=2f(3x).   &  &\notag
\end{xalignat*}

\exercise \item {\bf Graphs of standard \index{ elementary functions} elementary functions }
The following is a list of functions whose graphs you will need to become familiar with. 
For each, plot the graph for a suitable range of values of the independent variable, x. If there are additional parameters make plots for several values of the parameters. Label printed copies of the graphs with the following information: \\
\smallskip
i) the function which the graph illustrates \\
ii) the axes (x and y) \\
iii) the location of roots \\
iv) asymptotes \\

\begin{enumerate}
\item $f(x) = \sin(x)$.
\item $f(x) = a\sin(bx + c)$ 
\item $f(x) = a\cos(bx)+c$. 
\item $f(x) = \tan(x)$. 
\item $f(x) = \ln(x)$. 
\item $f(x) = e^{x}$. 
\item $f(x) = e^{-x}$. 
\item $f(x) = x$.
\item $f(x) = ax + b$. 
\item $f(x) = x^{2}$.
\item $f(x) = ax^{2} + bx + c$. 
\item $f(x) = x^{3}$.
\item $f(x) = ax^{3} + bx^{2} + cx + d$. 
%\newpage
\item $f(x) = \bigfrac{1}{x}$. 
\item $f(x) = \bigfrac{1}{x^{2}}$. 
\item $f(x) = \bigfrac{1}{x^{3}}$. 
\item $f(x) = |x|$. 
\item $f(x) = |x - 2|$. 
\end{enumerate}

\end{homework}
