1. Economics 214 Lecture 8 Introduction to Functions Cont.

2. Necessary & Sufficient Conditions Symbol  means “implies” P  Q means “if P then Q,” “P implies Q,” “P only if Q.” P is a sufficient condition for Q, for if P holds then if follows that Q holds. This expression also shows that Q is a necessary condition for P cause P can only hold if Q holds.

3. Example of Necessary and Sufficient Conditions f strictly concave  f concave

4. If and only if

5. Power Functions

6. Rules of Exponents

7. Power functions with p equal even integer F(0)=0 If k>0, then f(x) reaches a global minimum at x=0. If k<0, then f(x) reaches global maximum at x=0. These functions are symmetric about the vertical axis. They are strictly convex if k>0 or strictly concave if k<0.

8. Figure 2.13 Power Functions in Which the Exponent Is a Positive Even Number