227 2.3 Functions –4 2 6 –5 5 x y Range Domain y = 5 x – 1 0 Figure 24 Variations of the Definition of Function 1. A function is a relation in which, for each distinct value of the first component of the ordered pairs, there is exactly one value of the second component. 2. A function is a set of ordered pairs in which no first component is repeated. 3. A function is a rule or correspondence that assigns exactly one range value to each distinct domain value. Thus, the domain includes all real numbers except 1, written as the interval 1-∞, 12 ´11, ∞2. Values of y can be positive or negative, but never 0, because a fraction cannot equal 0 unless its numerator is 0. Therefore, the range is the interval 1-∞, 02 ´10, ∞2, as shown in Figure 24. S Now Try Exercises 37, 39, and 45. (e) Given any value of x in the domain of y = 5 x - 1 , we find y by subtracting 1 from x, and then dividing the result into 5. This process produces exactly one value of y for each value in the domain, so this equation defines a function. The domain of y = 5 x - 1 includes all real numbers except those that make the denominator 0. We find these numbers by setting the denominator equal to 0 and solving for x. x - 1 = 0 x = 1 Add 1. Function Notation When a function ƒ is defined with a rule or an equation using x and y for the independent and dependent variables, we say, “y is a function of x” to emphasize that y depends on x. We use the notation y =ƒ1x2, called function notation, to express this and read ƒ1x2 as “ƒ of x,” or “ƒ at x.” The letter ƒ is the name given to this function. For example, if y = 3x - 5, we can name the function ƒ and write ƒ1x2 = 3x - 5. Note that ƒ1x2 is just another name for the dependent variable y. For example, if y = ƒ1x2 = 3x - 5 and x = 2, then we find y, or ƒ122, by replacing x with 2. ƒ122 = 3 # 2 - 5 Let x = 2. ƒ122 = 1 Multiply, and then subtract. The statement “In the function ƒ, if x = 2, then y = 1” represents the ordered pair 12, 12 and is abbreviated with function notation as follows. ƒ122 = 1 The symbol ƒ122 is read “ƒ of 2” or “ƒ at 2.” LOOKING AHEAD TO CALCULUS One of the most important concepts in calculus, that of the limit of a function, is defined using function notation: lim xua ƒ1x2 =L (read “the limit of ƒ1x2 as x approaches a is equal to L”) means that the values of ƒ1x2 become as close as we wish to L when we choose values of x sufficiently close to a.
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