Economics 214

Lecture 4

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Graphing Inverse functionTo 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.

Graph of function and inverse function

Plot of f(x)=2/(1+x) and f-1(x)=2/x-1

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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.

Demand Function

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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.

Figure 2.8 A Function with Extreme Values

Average Rate of Change

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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 [xA,xB] if the secant line drawn on that interval lies wholly below the function.

Function is strictly convex in the interval [xA,xB] if the secant line drawn on that interval lies wholly above the function.

Figure 2.10 Strictly Concave and Strictly Convex Functions

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Figure 2.11 Concave and Convex Functions