Quiz 7 (Solution)

Date: MATH 1090-4 - Fall 2003

1. Match each of the following descriptions with the corresponding graph shown in Figure 1. No justification required!

   i.

   A rational function with vertical asymptotes.

   ii.

   A piecewise defined function.

   iii.

   A polynomial of degree $3$.

   Figure 1: Graphs for exercise 1

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   Graph (b) is the only graph with vertical asymptotes, so it corresponds to a rational function (with vertical asymptotes). Graph (a) is composed of two parts (a parabola and a line), so that it corresponds to a piecewise defined function. Graph (c) corresponds to a polynomial of odd degree, so it is the polynomial of degree $3$.

2. Identify any vertical asymptotes of $f(x)=\frac{2x+4}{x+1}$.

   Vertical asymptotes in rational functions appear when the denominator vanishes. In this case, $x+1=0 \ensuremath{\Rightarrow}\xspace x=-1$, so that the only vertical asymptote happens at $x=-1$.

3. Let

   $$A=\left[ \begin{array}{ccc} 1 & -1 & 2\\ 4 & 0 & 1 \end{array} \r... ...d{array} \right] \quad D=\left[ \begin{array}{c} 2\\ -7\\ 6 \end{array} \right]$$

   
   
   1. Write the size of each of the four matrices.

      $A$ is $2\times 3$; $B$ is $2\times 2$; $C$ is $2\times 2$; $D$ is $3\times 1$.

   2. Perform each of the following operations whenever possible. If it is not possible, say so.
      1. $A^t$.

         Each row of $A$ becomes a column:

         $$A^t = \left[ \begin{array}{cc} 1&4\\ -1&0\\ 2&1 \end{array} \right]$$

         
         

      2. $A+B$.

         It is not possible because $A$ and $B$ have different sizes.

      3. $B-C$.

         $$B-C = \left[ \begin{array}{cc} 0&1\\ 1&0 \end{array} \right] - \l... ...&0 \end{array} \right] = \left[ \begin{array}{cc} 1&0\\ 2&0 \end{array} \right]$$

         
         

      4. $D+D$.

         $$D+D = \left[ \begin{array}{c} 2\\ -7\\ 6 \end{array} \right] + \l... ... \end{array} \right] = \left[ \begin{array}{c} 4\\ -14\\ 12 \end{array} \right]$$