## 8.1. Pointwise Convergence

### Example 8.1.2 (a): Function Family Example

Consider the family of functions

*{ f*._{c}(x) } = { x^{2}- c }- What is the (natural) domain of the family?
- What is the (natural) parameter set of the family?
- List three particular members of the family.
- How many members does this family have?

1. For each fixed value of *c* the function
*{ f _{c}(x) } = { x^{2} - c }*
is a standard parabola, shifted up or down by

*c*. Thus, the natural domain for the family is

*(all real numbers).*

**R**2. The parabola *x ^{2}* 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(parabola shifted down by 1)_{-1}(x)f(standard parabola)_{0}(x)f(parabola shifted up by 1)_{1}(x)

Click on *Options* to plot additional members of this family.

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