Section 3.4 Library of Functions; Piecewise-defined Functions

Graph the Functions Listed in the Library of Functions

Constant Function xy=b

Identity Function xy=x

Square Function xy=x

Cube Function xy=x

Square Root Function xy=x 1/

Cube Root Function xy=x 1/

Reciprocal Function xy=1/x  4 4  2  1  1/2  1/3  1/4 0

xy=|x| Absolute Value Function

Greatest Integer Function x y=  x 

Piecewise-defined Functions

Use when x values satisfy condition n Use when x values satisfy condition 1 Sometimes we need more than one formula to specify a function algebraically. In this case the formula used to evaluate the function depends on the value of x. Piecewise Defined Functions

The following is a quick example of a piecewise defined function = 26.5 = 53.8 Use when x values are greater than 2 Use when x values are less than or equal to 2 Notice Example 1

Notice that the domain of f, in this case, is the set all real numbers. That is, Dom f = (– ,  ) The following is a quick example of a piecewise defined function Example 1

The percentage p (t) of buyers of new cars who used the Internet for research or purchase since 1997 is given by the following function.† (t = 0 represents 1997). Notice that the domain of p is the interval [0, 4]. That is, Dom p = [0, 4]. † The model is based on data through Source: J.D. Power Associates/The New York Times, January 25, 2000, p. C1 Example 2

This notation tells us that we use the first formula, 10t + 15, if 0  t < 1, or, t is in [0, 1) the second formula, 15t + 10, if 1  t  4, or, t is in [1,4] Example 2

Thus, for instance, p(0.5) = 10(0.5) + 15 = 20 Here we used the first formula since 0  0.5 < 1, or, equivalently, 0.5 is in [0, 1). p(2) = 15(2) + 10 = 40 We used the second formula since 1  2  4, or equivalently, 2 is in [1, 4]. p(4.1) is undefined p (t ) is only defined if 0  t  4. Example 2

The function f (x) is defined as Example 3

(a) (b) Example 4: Cost of Electricity

Therefore the function C (x) can be written as