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15 inverse functions if we have a well dened function

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1.5 INVERSE FUNCTIONS If we have a well-de±ned function y = f ( x ), de±ned over a given domain X and range Y , then, given one condition , an inverse function g will exist, such that x = g ( y ). Sometimes the function g will be denoted f –1 , “the inverse function of f ,” or simply “ f -inverse.” The condition is straightforward: y = f ( x ) must be a (monotonically) strictly increasing or strictly decreasing function of x . Figure M.1-5 provides some examples of functions with and without corresponding inverse functions. MATH MODULE 1: FUNCTIONS, GRAPHS, AND THE COORDINATE SYSTEM M1-7
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In Figure M.1-5(a), all three curves are linear (straight-line) graphs of functions, with y = f( x ) = b + mx, where b is the vertical intercept and m is the slope. Only (1) and (3), however, have inverse functions, with the form x = g ( y ) = ( y – b )/ m. In the case of the horizontal (constant) function (2), y = b , with m = 0, there is an in±nite number of val- ues of x corresponding to the only value that y takes on (the entire range of y is b ). Hence if we pick a value of x , say x 0 , then f( x 0 ) = y 0 = b , but there does not exist an inverse func- tion that will get us back directly to x 0 from y 0 = b . In Figure M.1-5(b), y = x 2 likewise has no inverse function, since every value of y in the range of the function (0, ) can be generated by two values of x : y = 4 can be gener- ated by x = +2 or by x = –2. If we restrict the domain of x to be non-negative, however, then an inverse function, x = g ( y ) = | Ï y w | = | y 0.5 |, does exist. In Figure M.1-5(c), there is no function y = f( x ), because each value of x generates two values for y , one positive and one negative. There does , however, exist a function, x = g ( y ) = y 2 , that assigns a value of x for each value of y. If we restrict the range of y to non- negative values, then y = f ( x ) = | Ï x w | = | x 0.5 |, (M.1.8) and the inverse function x = g ( y ) = y 2 also exists. Inverse functions have many uses in economics, but in this course one of their pri- mary uses is with demand and supply functions, where we ±nd both expressions relat- ing price and quantity useful, in different contexts. We need to be able to transform one functional form into its inverse, when the inverse exists and is useful. As an exercise, you may want to examine the Figures and equations throughout this Module, and iden- tify which are proper functions, and also which functions have an inverse. 1.6 FUNCTIONS OF SEVERAL VARIABLES Some of the most important functions you will study in the course are functions of sev- eral variables. Economists assume, for instance, that a consumer’s preferences can be represented by a utility function having the form U = f ( X 1 , X 2 , X 3 , . .. , X n ), (M.1.9) M1-8 MATH MODULE 1: FUNCTIONS, GRAPHS, AND THE COORDINATE SYSTEM y yy y = f ( x ) = b + mx xx b (a) = ( ) = 2 = 2 x = 2 (b) = g ( ) = 2 (c) m > 0 m = 0 m < 0 00 0 2+ 2 4 1 2 3 FIGURE M.1-5
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where U stands for the “utility” or level of satisfaction received by the consumer from consuming various combinations of the n goods X 1 to X n . Here we need to plug in a value for the level of consumption of each of the goods, or n numbers in total, to Fnd the level of satisfaction U.
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