1. Economics 214 Lecture 4

2. Special Property

3. Composite Function

4. Graphing Inverse function To graph the inverse function y=f -1 (x) given the function y=f(x), we make use of the fact that for any given ordered pair (a,b) associated with a one-to-one function there is an ordered pair (b,a) associated with the inverse function. To graph y=f -1 (x), we make use of the 45  line, which is the graph of the function y=x and passes through all points of the form (a,a). The graph of the function y=f -1 (x), is the reflection of the graph of f(x) across the line y=x.

5. Graph of function and inverse function

6. Common Example Inverse function Traditional demand function is written q=f(p), i.e. quantity is function of price. Traditionally we graph the inverse function, p=f -1 (q). Our Traditional Demand Curve shows maximum price people with pay to receive a given quantity.

7. Demand Function

8. Plot of our Demand function

9. Extreme Values The largest value of a function over its entire range is call its Global maximum. The smallest value over the entire range is called the Global minimum. Largest value within a small interval is a local maximum. Smallest value within a small interval is called a local minimum.

10. Figure 2.8 A Function with Extreme Values

11. Average Rate of Change

12. Average Rate of Change Linear Function

13. Average Rate of Change for Nonlinear function

14. Secant Line

15. Graph of Average Rate of Change for Nonlinear Function

16. Concavity and Convexity The concavity of a univariate function is reflect by the shape of it graph and its secant line. Function is strictly concave in the interval [x A,x B ] if the secant line drawn on that interval lies wholly below the function. Function is strictly convex in the interval [x A,x B ] if the secant line drawn on that interval lies wholly above the function.

17. Figure 2.10 Strictly Concave and Strictly Convex Functions

18. Strictly Concave

19. Strictly Convex

20. Convexity and secant line

21. Our Example

22. Concave

23. Convex

24. Figure 2.11 Concave and Convex Functions