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%  Properties of Functions: Domains, Intercepts, Symmetry           %
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\problemgrouptitle{Properties of Functions: Domains, Intercepts, Symmetry}

\begin{problemblock}[2]{Find the domain and the $x$- and $y$-intercepts (if there are any) of the given functions.}

\item $f(x)=x^3$
\answer{$D=\mathset{R}=(-\infty,\infty)$; $x$-int=0; $y$-int=0}

\item $h(x)=\sqrt{x-6}$
\answer{$D=[6,\infty)$; $x$-int$=6$; No $y$-int}

\item $f(x)=x^2+4x+4$
\answer{$D=\mathset{R}=(-\infty,\infty)$; $x$-int=-2; $y$-int=4}

\item $g(x)=x^4+4x^3-5x^2$
\answer{$D=\mathset{R}=(-\infty,\infty)$; $x$-int=-5, 0 ,1; $y$-int=0}

\item $f(x)=\displaystyle\frac{1}{x-1}$   
\answer{$D=\mathset{R}-\left\{1\right\}=(-\infty,1)\cup(1,\infty)$; No $x$-int; $y$-int$=-1$}

\item $h(x)=\displaystyle\frac{1}{x^2+4x+4}$
\answer{$D=\mathset{R}-\left\{-2\right\}=(-\infty,-2)\cup(-2,\infty)$; No $x$-int; $y$-int$\displaystyle=\frac{1}{4}$}

\item $p(x)=\displaystyle\sqrt{4-x^2}$
\answer{$D=[-2,2]$; $x$-int=-2, 2; $y$-int=2}

\item $\displaystyle f(x)=3x^2+5x+2$ \answer{$D=\mathset{R}=(-\infty,\infty)$; $x$-int$\displaystyle =-\frac{2}{3},-1$; $y$-int$=2$}

\item $\displaystyle g(x)=x^3+3x-4$
\answer{$D=\mathset{R}=(-\infty,\infty)$; $x$-int$=1$; $y$-int$=-4$}

\item $\displaystyle h(x)=\frac{x+5}{x+7}$
\answer{$D=\mathset{R}-\{-7\}=(-\infty,-7)\cup(-7,\infty)$; $x$-int$=-5$; $y$-int$\displaystyle =\frac{5}{7}$}

\item $\displaystyle f(x)=\frac{10}{2x^2-5x-3}$
\answer{$D=\mathset{R}-\left\{-1/2,3\right\}=(-\infty,-1/2)\cup(-1/2,3)\cup(3,\infty)$; No $x$-int; $y$-int$\displaystyle =-\frac{10}{3}$}

\item $\displaystyle g(x)=\sqrt{4-x}$
\answer{$D=\{x\in\mathset{R}|x\le4\}=(-\infty,4]$; $x$-int$=4$; $y$-int$=2$}

\item $\displaystyle p(x)=\sqrt{x^2-10}$
\answer{$D=\left(-\infty,-\sqrt{10}\right]\cup\left[\sqrt{10},\infty\right)$; $x$-int$=\pm\sqrt{10}$; No $y$-int}

\item $\displaystyle h(x)=\sqrt{\frac{x+6}{2x-3}}$
\answer{$D=\left(-\infty,-6\right]\cup\left(3/2,\infty\right)$; $x$-int$=-6$; No $y$-int}

\item $\displaystyle f(x)=\frac{\sqrt{x^2-10}}{x^2+10}$
\answer{$D=\left(-\infty,-\sqrt{10}\right]\cup\left[\sqrt{10},\infty\right)$; $x$-int$=\pm\sqrt{10}$; No $y$-int}

\item $\displaystyle f(x)=\frac{1}{\sqrt{x+2}}-\frac{1}{x}$
\answer{$D=(-2,0)\cup(0,\infty)$; $x$-int=2; No $y$-int}

\end{problemblock}

\begin{problemblock}[2]{Determine whether each of the given functions is even, odd or neither.}

\addtoanswerkey{\begin{multicols}{4}}
\item $f(x)=3x^2$
\answer{Even}

\item $g(x)=2x^3$
\answer{Odd}

\item $h(x)=3x^2+2x^3$
\answer{Neither}

\item $f(t)= -6t^3$
\answer{Odd}

\item $g(u)=u^3-u^2$
\answer{Neither}

\item $f(x)=x^3+x^5$
\answer{Odd}

\item $\displaystyle f(x)=\displaystyle\sqrt{4-x^2}$
\answer{Even}

\item $\displaystyle h(x)=\frac{x^2+1}{\displaystyle\sqrt{4-x^2}}$
\answer{Even}

\item $\displaystyle f(x)=5x^2+3$
\answer{Even}

\item $\displaystyle g(t)=4t^3+t$
\answer{Odd}

\item $\displaystyle h(z)=z^5+1$
\answer{Neither}

\item $\displaystyle f(t)=\sqrt{t^2+5}$
\answer{Even}

\item $\displaystyle g(x)=\frac{x}{x^4+3}$
\answer{Odd}

\item $\displaystyle h(u)=\frac{3+u^2}{u+1}$
\answer{Neither}

\item $\displaystyle p(x)=\frac{x^5}{x^3+x}$
\answer{Even}

\item $\displaystyle u(s)=\sqrt{\frac{s^6+4}{s^{10}+7}}$
\answer{Even}

\addtoanswerkey{\end{multicols}}
\end{problemblock}

\textbook{\clearpage}
\item Here is a graph of a function $f$:
\begin{center}
\resizebox{5cm}{!}{\asycode{%
import graph;
size(150,150);
real f(real x) {return -x^3;}
pair F(real x) {return (x,f(x));}
draw(graph(f,-1,1,operator ..),red+2);
xaxis("$x$",xmax=1.2,Arrow(HookHead,size=4),above=true);
yaxis("$y$",Arrow(HookHead,size=4),above=true);
}}
\end{center}
\begin{multicols}{2}
\raggedcolumns
\begin{enumerate}
\item Sketch the graph of $-f(x)$.
\item Sketch the graph of $f(-x)$.
\item Sketch the graph of $f(x+1)$.	\columnbreak
\item Sketch the graph of $f(x)+1$.
\item Is $f$ even, odd, or neither?
\end{enumerate}
\end{multicols}
\answer{
\begin{enumerate}
\begin{multicols}{2}
\raggedcolumns
\item ~

\resizebox{0.6\columnwidth}{!}{\asycode{%
import graph;
size(150,150);
real f(real x) {return x^3;}
pair F(real x) {return (x,f(x));}
draw(graph(f,-1,1,operator ..),red+2);
xaxis("$x$",xmax=1.2,Arrow(HookHead,size=4),above=true);
yaxis("$y$",Arrow(HookHead,size=4),above=true);
}}

\item ~

\resizebox{0.6\columnwidth}{!}{\asycode{%
import graph;
size(150,150);
real f(real x) {return x^3;}
pair F(real x) {return (x,f(x));}
draw(graph(f,-1,1,operator ..),red+2);
xaxis("$x$",xmax=1.2,Arrow(HookHead,size=4),above=true);
yaxis("$y$",Arrow(HookHead,size=4),above=true);
}}

\columnbreak
\item ~

\resizebox{0.6\columnwidth}{!}{\asycode{%
import graph;
size(150,150);
real f(real x) {return -(x+1)^3;}
pair F(real x) {return (x,f(x));}
draw(graph(f,-2,0.1,operator ..),red+2);
xaxis("$x$",xmax=0.7,Arrow(HookHead,size=4),above=true);
yaxis("$y$",Arrow(HookHead,size=4),above=true);
xtick("-1",-1);
}}

\item ~

\resizebox{0.6\columnwidth}{!}{\asycode{%
import graph;
size(150,150);
real f(real x) {return -x^3+1;}
pair F(real x) {return (x,f(x));}
draw(graph(f,-1,1,operator ..),red+2);
xaxis("$x$",xmax=1.2,Arrow(HookHead,size=4));
yaxis("$y$",Arrow(HookHead,size=4),above=true);
ytick("1",1);
}}

\item $f(-x)=-f(x)$, and so $f$ is odd.
\end{multicols}
\end{enumerate}
}
