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\problemgrouptitle{Equations and Functions Review}

\begin{problemblock}{Solve the given equations.}

\item $\displaystyle 2x^2+3x-2=0$
\answer{$x=1/2$, $x=-2$. Written as a solution set: $\{1/2,-2\}$}

\item $\displaystyle x^2+x-20=0$
\answer{$\{-5,4\}$}

\item $\displaystyle x^3+2x^2-x-2=0$
\answer{$\{-2,-1,1\}$}

\item $\displaystyle x^4-5x^2+4=0$
\answer{$\{\pm1,\pm2\}$}

\item $\displaystyle x^5-4x^3-x^2+4=0$
\answer{$\{-2, 1, 2\}$}

\end{problemblock}

\begin{problemblock}{Find the domain and the $x$- and $y$-intercepts of the given functions.}

\item $\displaystyle h(x)=\frac{2x-3}{5x+4}$
\answer{$\displaystyle D=\mathset{R}-\left\{-\frac{4}{5}\right\}$, $x$-int$\displaystyle=\frac{3}{2}$, $y$-int$\displaystyle=-\frac{3}{4}$}

\item $\displaystyle f(x)=\sqrt{2x^2-8}$
\answer{$D=\left(-\infty,-2\right]\cup\left[2,\infty\right)$, $x$-int=$2,-2$, $y$-int does not exist}

\item $\displaystyle g(x)=\sqrt{\frac{2x+1}{x+5}}$
\answer{$\displaystyle D=\left(-\infty,-5\right)\cup\left[-\frac{1}{2},\infty\right)$, $x$-int$\displaystyle=-\frac{1}{2}$, $y$-int$\displaystyle=\sqrt{\frac{1}{5}}$}

\item $\displaystyle p(x)=\frac{\sqrt{x+8}}{x-3}$
\answer{$D=\left[-8,3\right)\cup\left(3,\infty\right)$, $x$-int$=-8$, $y$-int$\displaystyle=-\frac{\sqrt{8}}{3}$}
\end{problemblock}

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

\addtoanswerkey{\begin{multicols}{4}}

\item $\displaystyle f(x)=\frac{x^4+3}{x^2+1}$
\answer{Even}

\item $\displaystyle g(t)=t^{2/3}$
\answer{Even}

\item $\displaystyle h(z)=z\sqrt{z^2+1}$
\answer{Odd}

\item $\displaystyle g(x)=\frac{x^3}{x^6+5}-x$
\answer{Odd}

\addtoanswerkey{\end{multicols}}

\end{problemblock}

\begin{problemblock}{Find the composite functions $f\circ g$ and $g\circ f$ and their domains.}

\item $\displaystyle f(x)=x^3+6$, $\displaystyle g(x)=x^{2/3}$
\answer{$\displaystyle f\circ g(x)=f(g(x))=x^{2}+6$ with $D=\mathset{R}$, $\displaystyle g\circ f(x)=g(f(x))=\sqrt[3]{\left(x^3+6\right)^2}$ with $D=\mathset{R}$}

\item $\displaystyle f(t)=\frac{2t+5}{t-4}$, $\displaystyle g(t)=t^2+3$
\answer{$\displaystyle f\circ g(t)=f(g(t))=\frac{2t^2+11}{t^2-1}$ with $D=\mathset{R}-\left\{1,-1\right\}$, $\displaystyle g\circ f(t)=g(f(t))=\frac{4t^2+20t+25}{\left(t-4\right)^2}+3$ with $D=\mathset{R}-\left\{4\right\}$}

\item $\displaystyle  f(x)=\sqrt{x-1}$, $\displaystyle g(x)=\frac{x+5}{x+3}$
\answer{$\displaystyle f\circ g(x)=f(g(x))=\sqrt{\frac{2}{x+3}}$ with $D=\left(-3,\infty\right)$, $\displaystyle g\circ f(x)=g(f(x))=\frac{\sqrt{x-1}+5}{\sqrt{x-1}+3}$ with $D=\left[1,\infty\right)$}
\end{problemblock}
