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Economics 214

Economics 214. Lecture 8 Introduction to Functions Cont. 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.

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Economics 214

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  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

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