To fill out a function's you will need to use test numbers before and after each of the function's and asymtopes
a). sign chart; values
b). rational equation; values
c). sign chart; zeroes
d). rational equation; zeroes

  C).  sign chart; zeroes

Step-by-step explanation:

A function potentially changes sign at each of its zeros and vertical asymptotes. So, to fill out a sign chart, you need to determine what the sign is on either side of each of these points. You can do that using test numbers, or you can do it by understanding the nature of the zero or asymptote.

Examples:

f1(x) = (x -3) . . . . changes sign at the zero x=3. Is positive for x > 3, negative for x < 3.

f2(x) = (x -4)^2 . . . . does not change sign at the zero x=4. It is positive for any x ≠ 4. This will be true for any even-degree binomial factor.

f3(x) = 1/(x+2) . . . . has a vertical asymptote at x=-2. It changes sign there because the denominator changes sign there.

f4(x) = 1/(x+3)^2 . . . . has a vertical asymptote at x=-3. It does not change sign there because the denominator is of even degree and does not change sign there.

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