8.1. Pointwise Convergence

Example 8.1.2 (a): Function Family Example

Consider the family of functions { fc(x) } = { x2 - c }.
  1. What is the (natural) domain of the family?
  2. What is the (natural) parameter set of the family?
  3. List three particular members of the family.
  4. How many members does this family have?

1. For each fixed value of c the function { fc(x) } = { x2 - c } is a standard parabola, shifted up or down by c. Thus, the natural domain for the family is R (all real numbers).

2. The parabola x2 can be shifted up or down by any amount. Thus, the natural parameter set for our family is also R.

3. The applet below shows:

  • f-1(x) (parabola shifted down by 1)
  • f0(x) (standard parabola)
  • f1(x) (parabola shifted up by 1)

Click on Options to plot additional members of this family.

4. Since the parameter set is the set of all real numbers, the family { fc(x) } contains uncountably many member functions.

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