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CHAPTER 1 FUNCTIONS AND LIMITS version: 1.1 Animation 1.1: Function Machine Source and credit: eLearn.Punjab

1. Quadratic Equations eLearn.Punjab 1. Quadratic Equations eLearn.Punjab 1. Functions and Limits eLearn.Punjab 1. Functions and Limits eLearn.Punjab 4 version: 1.1 version: 1.1 5 (iv) f(1/x) = (1/x) 3 - 2(1/x) 2 + 4 (1/x) - 1 = 1 x 3 - 2 x 2 + 4 x - 1, x ≠ 0 Example 2: Let f(x) = x 2 . Find the domain and range of f. Solution: f(x) is deined for every real number x. Further for every real number x, f(x) = x 2 is a non-negative real number. So Domain f = Set of all real numbers. Range f = Set of all non-negative real numbers. Example 3: Let f(x) = x x 2 - 4 . Find the domain and range of f. Solution: At x = 2 and x = -2, f(x) = x x 2 - 4 is not deined. So Domain f = Set of all real numbers except -2 and 2 . Range f = Set of all real numbers. Example 4: Let f(x) = x 2 - 9 . Find the domain and range of f. Solution: We see that if x is in the interval -3 < x < 3, a square root of a negative number is obtained. Hence no real number y = x 2 - 9 exists. So Domain f = { x d R : |x| 8 3 } = (-T, -3] j [3, + T) Range f = set of all positive real numbers = (0, + T) 1.1.4 Graphs of Algebraic functions If f is a real-valued function of real numbers, then the graph of f in the xy-plane is deined to be the graph of the equation y = f(x). The graph of a function f is the set of points {(x, y)| y = f(x)} , x is in the domain of f in the Cartesian plane for which (x, y) is an ordered pair of f. The graph provides a visual technique for determining whether the set of points represents a function or not. If a vertical line intersects a graph in more than one point, it is not the graph of a function. Explanation is given in the igure. Method to draw the graph: To draw the graph of y = f(x), we give arbitrary values of our choice to x and ind the corresponding values of y. In this way we get ordered pairs (x1 , y1 ) , (x2 , y2 ), (x3 , y3 ) etc. These ordered pairs represent points of the graph in the Cartesian plane. We add these points and join them together to get the graph of the function. Example 5: Find the domain and range of the function f(x) = x 2 + 1 and draw its graph. Solution: Here y = f(x) = x 2 + 1 We see that f(x) = x 2 +1 is deined for every real number. Further, for every real number x, y = f(x) = x 2 + 1 is a non-negative real number. Hence Domain f = set of all real numbers and Range f = set of all non-negative real numbers except the points 0 7 y < 1. For graph of f(x) = x 2 +1, we assign some values to x from its domain and ind the corresponding values in the range f as shown in the table: x -3 -2 -1 0 1 2 3 y = f(x) 10 5 2 1 2 5 10 Plotting the points (x, y) and joining them with a smooth curve, we get the graph of the function f(x) = x 2 + 1, which is shown in the igure.
1. Quadratic Equations eLearn.Punjab 1. Quadratic Equations eLearn.Punjab 1. Functions and Limits eLearn.Punjab 1. Functions and Limits eLearn.Punjab 6 version: 1.1 version: 1.1 7 1.1.5 Graph of Functions Deined Piece-wise. When the function f is deined by two rules, we draw the graphs of two functions as explained in the following example: Example 7: Find the domain and range of the function deined by: f(x) = x when 0 7x 71 x - 1 when 1 < x 72 also draw its graph. [ Solution: Here domain f = [0, 1] j [1, 2] = [0, 2]. This function is composed of the following two functions: (i) f(x) = x when 0 7 x 7 1 (ii) f(x) = x - 1 , when 1 < x 7 2 To ind th table of values of x and y = f(x) in each case, we take suitable values to x in the domain f. Thus Table for y = f(x) = x Table of y = f(x) = x - 1: x 0 0.5 0.8 1 x 1.1 1.5 1.8 2 y = f(x) 0 0.5 0.8 1 y = f(x) 0.1 0.5 0.8 1 Plotting the points (x, y) and joining them we get two straight lines as shown in the igure. This is the graph of the given function. 1.2 TYPES OF FUNCTIONS Some important types of functions are given below: 1.2.1 Algebraic Functions Algebraic functions are those functions which are deined by algebraic expressions. We classify algebraic functions as follows: (i) Polynomial Function A function P of the form P(x) = an x n + an-1 x n-1 + an-2 x n-2 + .... + a2 x 2 + a1 x + a0 for all x, where the coeicient an , an-1 , an-2 , .... , a2 , a1 , a0 are real numbers and the exponents are non-negative integers, is called a polynomial function. The domain and range of P(x) are, in general, subsets of real numbers. If an ≠ 0 , then P(x) is called a polynomial function of degree n and an is the leading coeicient of P(x) . For example, P(x) = 2x 4 - 3x 3 + 2x - 1 is a polynomial function of degree 4 with leading coeicient 2. (ii) Linear Function If the degree of a polynomial function is 1, then it is called a linear function. A linear function is of the form: f(x) = ax + b (a ≠ 0), a, b are real numbers. For example f(x) = 3x + 4 or y = 3x + 4 is a linear function. Its domain and range are the set of real numbers. (iii) Identity Function For any set X, a function I : X " X of the form I(x) = x " x d X , is called an identity function. Its domain and range is the set X itself. In particular, if X = R , then I(x) = x , for all x d R , is the identity function. (iv) Constant function Let X and Y be sets of real numbers. A function C : X " Y deined by C(x) = a , " x d X , a d Y and ixed, is called a constant function. For example, C : R " R deined by C(x) = 2, " x d R is a constant function. (v) Rational Function A function R(x) of the form P(x) Q(x) , where both P(x) and Q(x) are polynomial functions and Q(x) ≠ 0, is called a rational function. The domain of a rational function R(x) is the set of all real numbers x for which Q(x) ≠ 0. 1.2.2 Trigonometric Functions We denote and deine trigonometric functions as follows: