In this problem, we will examine the short-run behavior of the following functions. As you complete the chart, look carefully for patterns.

a) $f(x)= \dfrac{x-5}{x + 7}$

b) $g(x)= \dfrac{\left(x+3\right)\left(x-6\right)}{x^2-1}$

c) $h(x)= \dfrac{2\left(x + 1\right)\left(x-7\right)}{x + 1}$

d) $\ell(x)= \dfrac{\left(x + 2.8\right)\left(x + 6\right)}{\left(x+1\right)\left(x + 6 \right)}$

e) $m(x)= \dfrac{x^2-6x+8}{x-2}$

Complete the following table using the formulas for each of the above functions and any additional information you can obtain from the graph of each function. For this question, consider holes and vertical asymptotes to be undefined points.

Function features
functionnumber of (distinct) zerosnumber of undefined points
$f(x)$
$g(x)$
$h(x)$
$\ell(x)$
$m(x)$

b) List the roots (in form $x = r$) in order from smallest to largest. Also, list the undefined points of the function from smallest to largest. Finally, list the vertical asymptote(s). If there are more blanks provided than needed for a given feature, enter NONE in all remaining, unneeded blanks. The information for function $f(x)$ has been completed as an example.

$f(x)$:
Zeros: $x = 4$, $x =$NONE
Holes: $x =$ NONE, $x=$ NONE
Vertical Asymptotes: $x =-6$, $x=$ NONE

$g(x)$:
Zeros: $x =$ , $x =$
Holes: $x =$ , $x=$
Vertical Asymptotes: $x =$ , $x=$

$h(x)$:
Zeros: $x =$ , $x =$
Holes: $x =$ , $x=$
Vertical Asymptotes: $x =$ , $x=$

$\ell(x)$:
Zeros: $x =$ , $x =$
Holes: $x =$ , $x=$
Vertical Asymptotes: $x =$ , $x=$

$m(x)$:
Zeros: $x =$ , $x =$
Holes: $x =$ , $x=$
Vertical Asymptotes: $x =$ , $x=$