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9.6 - Matching game
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<h2>
9.6 - The matching game
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<p>Match each function with its graph.  Give reasons for your choices.  (Your reasons might include arguments involving <i>traces</i> of $x$, $y$ or $z$)

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<td>1)&nbsp;$f(x,y)=\frac{1}{x+1}+\sin y$. 
<p class="ustme">the fraction blows up as $x\to-1$.  Otherwise, if $x=k$, then $z\propto C+\sin(y)$.
<br>$\Rightarrow$ III
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<td>2)&nbsp;$f(x,y)=\sqrt{4-x^2-y^2}$
<p class="ustme">Horizontal traces ($z=k$) are circles.  Impossible for $z$ to be greater than $\sqrt 4=2$.<br>$\Rightarrow$ VI
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<tr valign="top"><td>3)&nbsp;$f(x,y)= \cos(x+y^2)$
<p class="ustme">If $y^2=k$, then $z\propto \cos(x+k)$. Also, the $\cos(\text{ })$ function never gets above 1 or below -1, so $-1< z < +1$ as well.<br>
$\Rightarrow$ V
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<td>4)&nbsp;$f(x,y)=\ln(x^2+y^2+1)$
<p class="ustme">Lowest possible value occurs when $x=y=0$:  $f(0,0)=\ln 1=0$.
Horizontal traces are circles.<br>
$\Rightarrow$ I</td>
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<td>5) $f(x,y)=x^2\sqrt y$
<p class="ustme">If $y=k$, $z\propto x^2$.  If $x=k$, $z\propto \sqrt y$.  Undefined for $y\lt 0$.<br>
$\Rightarrow$ IV</td>
<td>6) $f(x,y)=x^3y$
<p class="ustme">When $y=k$, $z\propto x^3$.  When $x=k$, $z\propto x$.<br>
$\Rightarrow$ II</td>
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