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Economics 214

Economics 214. Lecture 6 Properties of Functions. Increasing & Decreasing Functions. Figure 2.6 Increasing Functions and Decreasing Functions. Example functions. Monotonic Functions. Monotonic if it increasing or if it is decreasing.

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Economics 214

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  1. Economics 214 Lecture 6 Properties of Functions

  2. Increasing & Decreasing Functions

  3. Figure 2.6 Increasing Functions and Decreasing Functions

  4. Example functions

  5. Monotonic Functions • Monotonic if it increasing or if it is decreasing. • Strictly monotonic if it is strictly increasing or if it is strictly decreasing. • Non-monotonic if it is strictly increasing over some interval and strictly decreasing over another interval. • Strictly monotonic functions are one-to-one functions.

  6. One-to-One Function • A function f(x) is one-to-one if for any two values of the argument x1 and x2, f(x1)= f(x2) implies x1=x2. • An one-to-one function ha an inverse function. • The inverse of the function y=f(x) is written y=f -1(x).

  7. Inverse Function

  8. Example

  9. Graph of example

  10. Graph of inverse function

  11. Special Property

  12. Composite Function

  13. 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.

  14. Graph of function and inverse function

  15. 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.

  16. Demand Function

  17. Plot of our Demand function

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