Functions - CHAPTER 4 Functions Functions, and the language...

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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 1.6-8 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 x-axis once (or not at all); a quadratic function crosses the x-axis twice, or touches it once, or not at all. The graph of a polynomial of degree n crosses the x-axis 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 x-axis 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.

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Functions - CHAPTER 4 Functions Functions, and the language...

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