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Unformatted text preview: CHAPTER 4 Functions Functions, and the language of functions, are widely used in economics. Linear and quadratic functions were discussed in Chapter 2. Now we in troduce other useful functions (including exponential and logarithmic functions) and concepts (such as inverse functions and functions of several variables ). Economic applications are supply and demand functions , utility functions and production functions . — ./ — 1. Function Notation, and Some Common Functions We have already encountered some functions in Chapter 2. For example: y = 5 x 8 Here y is a function of x (or in other words, y depends on x ). C = 3 + 2 q 2 Here, a firm’s total cost, C , of producing output is a function of the quantity of output, q , that it produces. To emphasize that y is a function of x , and that C is a function of q , we often write: y ( x ) = 5 x 8 and C ( q ) = 3 + 2 q 2 Also, for general functions it is common to use the letter f rather than y : f ( x ) = 5 x 8 Here, x is called the argument of the function. In general, we can think of a function f ( x ) as a “black box” which takes x as an input, and produces an output f ( x ): x . . . . . . . . . . . f ( x ) . . . . . . . . . . . f For each input value, there is a unique output value. 57 58 4. FUNCTIONS Examples 1.1 : For the function f ( x ) = 5 x 8 (i) Evaluate f (3) f (3) = 15 8 = 7 (ii) Evaluate f (0) f (0) = 0 8 = 8 (iii) Solve the equation f ( x ) = 0 f ( x ) = 0 = ⇒ 5 x 8 = 0 = ⇒ x = 1 . 6 (iv) Hence sketch the graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f ( x ) x 1.68 Exercises 4.1 : Using Function Notation (1) (1) For the function f ( x ) = 9 2 x , evaluate f (2) and f ( 4). (2) Solve the equation g ( x ) = 0, where g ( x ) = 5 10 x +1 . (3) If a firm has cost function C ( q ) = q 3 5 q , what are its costs of producing 4 units of output? 1.1. Polynomials Polynomials were introduced in Chapter 1. They are functions of the form: f ( x ) = a n x n + a n 1 x n +1 + ... + a Examples 1.2 : (i) g ( x ) = 5 x 3 2 x + 1 is a polynomial of degree 3 (ii) h ( x ) = 5 + x 4 x 9 + x 2 is a polynomial of degree 9 A linear function is a polynomial of degree 1, and a quadratic function is a polynomial of degree 2. We already know what their graphs look like (see Chapter 2); a linear function crosses the xaxis once (or not at all); a quadratic function crosses the xaxis twice, or touches it once, or not at all. The graph of a polynomial of degree n crosses the xaxis up to n times. 4. FUNCTIONS 59 Exercises 4.2 : Polynomials (1) (1) For the polynomial function f ( x ) = x 3 3 x 2 + 2 x : (a) Factorise the function and hence show that it crosses the xaxis at x = 0, x = 1 and x = 2....
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This note was uploaded on 04/12/2010 for the course ECON DEAM taught by Professor Vines during the Spring '10 term at Oxford University.
 Spring '10
 Vines
 Economics

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